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AMERICAN MACHINIST GEAR BOOK 



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American Machinist 
Gear Book 



Simplified Tables and Formulas 
for Designing, and Practical Points in 
Cutting All Commercial Types of Gears 



By Charles H. Logue 

Formerly Associate Editor American Machinist, 
Formerly Mechanical Engineer R. D. Nuttall Co. 



Revised Edition 

Reginald Trautschold, M. E. 
Editor 



McGRAW-HILL BOOK COMPANY, Inc. 
239 WEST 39TH STREET. NEW YORK 



LONDON: HILL PUBLISHING CO., Lin. 
(*) & 8 DOUVKUI1Q ST., E. C. 



Copyright, 191 8, by the 
McGraw-Hill Book Company, Inc. 



Copyright, 1910, by the American Machinist 
Copyright, 1915, by the American Machinist 



4$ 



stf lb 1919 




\ r i-m^ 



©CJ.A53 771 






PREFACE TO REVISED EDITION 

The first edition of this work, published some five years ago, met with 
gratifying success as it filled a widespread demand for reliable information 
on the subject of gearing. The last few years have added much to our 
knowledge concerning certain types of gears however, so that the present 
is a fitting time to revise and bring this work up to date. In doing so, the 
original plan of the work has been followed and much of its text retained 
intact, while considerable new matter has been added. The sections on 
bevel, worm, helical, and skew bevel gears have been rewritten. The 
section on pattern work and molding has been omitted and one touching 
upon the cost of standard gears substituted therefor. 

Reginald Trautschold. 

September i, 191 5. 



PREFACE 

This book has been written to fill a pressing want; to give practical data for 
cutting, molding, and designing all commercial types, and to present these 
subjects in the plainest possible manner by the use of simple rules, diagrams, 
and tables arranged for ready reference. In other words, to make it a book 
for " the man behind the machine, " who, when he desires information on a 
subject, wants it accurate and wants it quick, without dropping his work to 
make a general study of the subject. At the same time a general outline of 
the underlying principles is given for the student, who desires to know not 
only how it is made, but what is made. Controversies and doubtful theories 
are avoided. Tables and formulas commonly accepted are given without com- 
ment. A great deal of this matter has previously been published in the col- 
umns of the American Machinist, but is revised to make the subject more 
complete. Credit is given in all cases when the author is known; there may 
be cases however, where record of the original source of information has been 
lost, as is often the case when data are in daily use and the authority is ob- 
scure. Obviously in such cases the author's name cannot be given. 

Charles H. Logue. 
May i, 1910. 



CONTENTS 

SECTION PAGE 

I. Tooth Pabts i 

II. Spur Gear Calculations 45 

III. Speeds and Powers 49 

IV. Gear Proportions and Details op Design no 

V. Bevel Gears ' 139 

VI. Worm Gears 163 

VII. Helical and Herringbone Gears 204 

VIII. Spiral Gears 225 

IX. Skew Bevel Gears 245 

X. Intermittent Gears 254 

XI. Elliptical Gears 265 

XII. Epicyclic Gear Trains 272 

XIII. Friction Gears 288 

XIV. Odd Gearing 308 

XV. Costs 317 

XVI. Suggestions for Ordering Gears 325 

XVII Practical Points in Gear Cutting 338 

Index 341 



SECTION I 

Tooth gearing furnishes an efficient and simple means of transmitting power 
at a constant speed ratio, making it possible to time the movements of machine 
parts positively. 

Owing to refinements in the tooth form, the introduction of generating 
machines and facilities to cut gears of the largest size accurately, loads may 
now be transmitted at speeds which a comparatively short time ago were 
considered prohibitive. The need for gears that would answer the exacting 
requirements of automobile construction has done much to bring this about. 
Designing automobile gears, however, is a case of fitting the gears to the 
machine; it is a question of securing material that will stand the strain; the 
gear dimensions are practically self determined ; however, this is not the only 
kind of gearing that has been designed after this fashion. 

We have an excellent formula for the strength of gear teeth, but it contains 
a variable factor — the allowance to be made on account of impact — concerning 
which very little is known. The most important question of all, that of wear, 
has heretofore been left practically untouched. The best data obtainable 
has been given. Few records have been kept of actual performances, and 
nothing whatever has been found relative to the abrasion of different materials 
in tooth contact. 

The various ways in which gears are mounted is responsible for the apparent 
contradiction of what few data are at hand, as a gear driver which is entirely 
satisfactory on one machine will be worthless on another at the same load 
and the same speed. 

The circumferential speed that may be allowed for gears of different types 
is another neglected factor, and last, but not least, what do we know of gear 
efficiency? In fact the most important information relative to gear trans- 
missions has been entirely a matter of guesswork. It is hardly to be assumed 
that this will ever be reduced to an exact equation, but there should be some 
basis from which to form our conclusions. 

Gears may be roughly divided into three general classes: Gears connecting 
parallel shafts; gears connecting shafts at any angle in the same plane; gears 
connecting shafts at any angle not in the same plane. 

In the first class are included spur, helical, herringbone, and internal gears. 
The second class covers bevel gears only. The third class includes worm, 
spiral and skew bevel gears. 



2 AMERICAN MACHINIST GEAR BOOK 

Gears connecting parallel shafts are the most efficient, and from a point of 
efficiency may be graded into herringbone, internal, spur, and, lastly, helical 
gears. 

The efficiency of the second class, bevel gears, varies with the shaft angle, 
increasing as the angle approaches zero. 

As a general thing the third class should be avoided wherever possible, 
although worm gears have their peculiar uses; for instance, where a quiet, 
self-locking drive is required without reference to the loss of power. 

Spiral gears are employed where the load is light and the gear ratio is low, 
say under 10 to i; worm gears are often employed for low ratios down 
to i to i, but are extremely difficult to cut and therefore expensive. When 
the worm is made much coarser than quadruple thread there is generally 
trouble. 

Skew bevel gears are used where the distance between the shafts is not 
great enough to employ worm or spiral gears. Skew bevel gears are simpler, 
and easier to cut than has been generally supposed, but are still things to 
avoid. 

These three general classifications are commercially subdivided as follows: 



KIND 



Spur 
Bevel 

Helical 

Herringbone 

Spiral 

Worm 

Skew Bevel 

Internal 

Elliptical 

Irregular 

Intermittent 

Friction 



RELATION OF AXES 



Parallel 
Intersecting at any angle in 

the same plane 

Parallel 

Parallel 

At any angle not in same 

plane 

At any angle not in same 

plane 

At any angle not in same 

plane 

Parallel or at any angle in 

same plane 

Parallel or at any angle in 

same plane 

Parallel or at right angles in 

same plane 

Parallel or at right angles in 

same plane 



PITCH SURFACES 



Parallel or at any angle in 
same plane 



Cylinders 

Cones 

Cylinders 

Cylinders 

Cylinders 

Cylinders 

Hyperboloids 

Cylinders or cones 

Elliptical cylinders or 
elliptical cones 
Any 

Cylinders 



Cylinders or cones 



NOTES 



Double Helical 
For small ratios 
For large ratios 

Where shaft centers 
are close 

With teeth cut on in- 
ner surface 



Irregular pitch lines 
To give driven gear a 
period, or periods of 
rest during one rev- 
olution of driver 
Contact surfaces rep- 
resenting the pitch 
surfaces of a toothed 
gear 



Commercial Classification of Gears 



TOOTH PARTS 



TOOTH PARTS 

Fixed axes are connected by imaginary pitch surfaces, which roll upon each 
other and transmit uniform motion without slipping. The object in toothed 
gearing is to provide these imaginary surfaces with teeth, the action of which 
will make the uniform motion of the pitch surfaces positive; not depending 
upon friction produced by lateral pressure as in friction gears, which are an ex- 
cellent representation of pitch surfaces. 

If the teeth are not so formed that this condition is fulfilled the movement 
of the driven gear will be made up of accelerations and retardations which 
will not only absorb a large percentage of the power but disintegrate the 
material of which the gear is constructed 
and seriously affect the operation of the 
machine. Tool marks on planer and bor- 
ing mill work corresponding to the teeth 
in the driven gear may be traced directly 
to this. 

There is but one form of tooth in 
common use — the involute; the cycloidal 
form has practically disappeared. For 
a thorough understanding of tooth con- 
tact, however, it must be included. 




CYCLOIDAL 



FIG. I. GENERATING THE CYCLOIDAL 
TOOTH. 



Generated by rolling a circle above and 
below the pitch circle of gear; a point on its circumference describing the 
tooth outline. See Fig. i. 

INVOLUTE 

Generated by rolling a straight line on the base circle of gear, any point on 
this line describing the involute curve. See Fig. 2. The same result is obtained 
by unwinding a string from the base circle. See Fig. 3. 

OCTOID 

Conjugated by a tool representing a flat sided crown gear tooth; a modi- 
fication of the involute. Used only on bevel gear generating machines. See 

Fig. 33- 

THE CYCLOID 

An illustration of the manner in which the cycloidal tooth is generated is 
illustrated by Fig. 1; the wheel A being the pitch circle and B and B' the 
describing circles which are of the same diameter. The point C will describe 
the face of the tooth as the circle B is rolled on the pitch circle, and the Hank 
of the tooth as the circle B' is rolled inside the pitch circle. In other words, 



AMERICAN MACHINIST GEAR BOOK 



the exterior cycloid is formed by rolling the describing circle on the outside 
of the pitch circle, this exterior cycloid engaging the interior cycloid, which 
is formed by rolling the describing circle on the inside of the pitch circle. 

The describing circle is commonly made equal to the pitch radius of a 15- 
tooth pinion of the same pitch as the gear being drawn. 

According to J. Howard Cromwell: "Roomer, a celebrated Danish astrono- 
mer, is said to have been the first to demonstrate the value of these curves for 
tooth profiles." But De la Hire is credited with demonstrating that it was 




Base Circle 



FIG. 2. THE INVOLUTE GEN- 
ERATED BY A STRAIGHT LINE. 




FIG. 3. THE INVOLUTE GEN- 
ERATED BY A STRING. 



possible to form both the face and flanks of any number of gears with the same 
describing circle. 

The pressure angle of the teeth is not constant in one direction, but varies 
from zero at the pitch point to about 22 degrees at the end of the contact 
with a rack tooth. The contact points of all the teeth engaged intersect the 
line of action, which is a segment of the describing circle drawn from the line 
of centers. See Fig. 34. 

Wilfred Lewis has said: "The practical consideration of cost demands the 
formation of gear teeth upon some interchangeable system. 

"The cycloidal system cannot compete with the involute, because its cutters 
are formed with greater difficulty and less accuracy, and a further expense is 
entailed by the necessity for more accurate center distances. Cycloidal teeth 
must not only be accurately spaced and shaped but their wheel centers must 
be fixed with equal care to obtain satisfactory results. Cut gears are not only 
more expensive in this system, but also when patterns are made for castings 



TOOTH PARTS 5 

the double curved faces require far more time and care in chiseling. An involute 
tooth can be shaped with a straight-edged tool, such as a chisel or a plane, 
while the flanks of cycloidal teeth require special tools, approximating in cur- 
vature the outline desired. It is, therefore, hardly necessary to argue any 
further against the cycloidal gear teeth, which have been declining in popu- 
larity for many years, and the question now to be considered is the angle of 
obliquity most desirable for interchangeable involute teeth." 

In this same connection George B. Grant, of the Philadelphia Gear Works, 
wrote: " There is no more need of two different kinds of tooth curves for 
gears of the same pitch than there is need of two different threads for standard 
screws, or of two different coins of the same value, and the cycloidal tooth 
would never be missed if it were dropped altogether. But it was first in the 
field, is simple in theory, is easily drawn, has the recommendation of many 
well-meaning teachers and holds its position by means of 'human inertia,' or 
the natural reluctance of the average human mind to adopt a change, partic- 
ularly a change for the better. " 

THE INVOLUTE 

The pressure on the teeth of involute gears is constantly in the direction of 
the line of action. The line of action is drawn through the pitch point at an 
angle from the horizontal equal to the angle of obliquity. All contact between 
the teeth is along this line. The base circle is drawn inside the pitch circle and 
tangent to the line of action. 

The action of a pair of involute gears is the same as if their base circles were 
connected by a cross belt; the point at which the belt crosses being the pitch 
point P; the straight portion of the belt not touching the base circles represent- 
ing the lines of action. See Fig. 4. At the pitch point the velocities of both 
gears are equal. To show that the involute is but a limiting case of the cy- 
cloidal system, consider the describing line as a curve of infinite radius, which 
is rolled upon the pitch circle. As this describing line cannot be rolled inside 
the pitch circle to form the interior cycloid that will engage the exterior cycloid 
formed by rolling the describing line outside the pitch circle of the mating 
gear, the pitch .circles upon which the cycloids are formed must be separated 
so as to allow the exterior cycloids to engage each other. The original pitch 
circles becoming the base circles. See Fig. 5. 

The distance between the pitch circle and the base circle, and therefore, the 
angle of obliquity, depends upon the proportionate length of tooth to be used 
and the smallest number of teeth in the system. To obtain contact for the 
full length of the tooth, the base circle must fall below the lowest point reached 



AMERICAN MACHINIST GEAR BOOK 



by the teeth of the mating gear. Below the base line there can be no contact 
of any value. 

There is such a difference between the largest possible gear and the rack 
that it is at first a little difficult to see the application of the methods used to 
describe the involute to the rack 
tooth. 

As the diameter of the gear is in- 
creased, the radii used to draw the 



Pitch Circle 





FIG. 4. THE ACTION OF INVOLUTE TEETH 
ILLUSTRATED BY A CROSSED BELT CON- 
NECTING THE BASE CIRCLES. 



FIG. 5. SEPARATING THE PITCH CIRCLES 
TO ALLOW THE EXTERIOR CYCLOIDS TO 
ENGAGE. 



involute curve are lengthened, and the teeth have less curvature. Until finally, 
when the radius of the pitch circle is of infinite length, the tooth radii are also 
infinite, and the involute is a straight line, drawn at right angles to the line 
of action. 

The theoretical rack tooth, therefore, has perfectly flat sides, each side being 
inclined toward the center of the tooth to an angle equaling the angle of ob- 
liquity. See Fig. 6, 



TOOTH PARTS 



ORIGIN OF THE INVOLUTE TOOTH 

The origin of the involute curve as applied to the teeth of gears is credited 
to De la Hire, a French scientist, a complete description and explanation of 
its use being published about 1694 in Paris. The first English translation of 
this work was published in London in 1696 by Mandy.* Professor Robinson, 
of Edinburgh, later describes this theory, references being made to his work in 
"An Essay on Teeth of Wheels," by Robertson Buchannan, edited by Peter 
Nicholson and published in 1808. In this essay the involute as applied to the 
teeth of gears is fully described, Fig. 7 being a copy of a cut used therein for 
illustration. That the principal ad- 
vantage of the involute system was 
then well understood will be shown 
in the following paragraph, referring 
to Fig. 7 : 





FIG. 6. THE INVOLUTE RACK TOOTH. 



FIG. 7. ACTION OF THE INVOLUTE TOOTH. 



"It is obvious that these teeth will work both before and after passing the 
line of centers, they will work with equal truth, whether pitched deep or shal- 
low, a quality peculiar to them and of very great importance." 

The theory of the involute gear tooth is also described by Sir David Brew- 
ster, Dr. Thomas Young, Mr. Thomas Reid and others. 

Professor Robert Willis, gives a very complete description of this form of 
tooth in his "Principles of Mechanism," 1841. Up to this period the involute 
tooth was not seriously considered, the cyclodial being the favorite. The in- 

* However, the origin of the involute gear tooth is surrounded by mystery, no two authorities 
agreeing upon the subject. According to Robert Willis, in his "Principles of Mechanism," 
the involute was first suggested for this purpose by Euler, in his second paper on the Teeth 
of Wheels. N.C. Petr XI. 209. 




8 AMERICAN MACHINIST GEAR BOOK 

volute tooth was objected to on account of the great thrust supposed to be put 
on the bearings by the oblique action of the teeth. 

In an 1842 edition of M. Camus' work, "A treatise on the Teeth of Wheels," 
edited by John I. Hawkins, a series of experiments with wooden models was 
made to demonstrate the actual thrust occasioned by different angles of ob- 
liquity. The result of these experiments is given as follows: 

" These experiments, tried with the most scrupulous attention to every cir- 
cumstance that might affect their result, elicit this important fact — that the 
teeth of wheels in which the tangent of the surfaces in contact makes a less 
angle than 20 degrees with the line of centers, possess no tendency to cause a 

separation of their axes: consequently, 
there can be no strain thrown upon the 
bearings by such an obliquity of tooth. " 
J. Howard Cromwell, in his treatise 
on Tooth Gearing, 1901, says: "Such an 
obliquity as 20 degrees must, unless 
counteracted by an opposing force, tend 
to separate the axes; and, as suggested 
fig. 8. the molding process. by Mr. Hawkins, this opposing force is 

most probably the friction between the 
teeth, which would tend to drag the axes together with as much force as 
that tending to separate them." 

That the involute system is closely connected to the cycloidal system is 
shown by Dr. Brewster in his reference to De la Hire's work. 

"De la Hire considered the involute of a circle as the last of the exterior 
epicycloids; which it may be proved to be, if we consider the generating 
straight line (see Fig. 2) as a curve of infinite radius. " 

The 143^2 degree angle of obliquity, as proposed by Professor Robert Willis 
in his "Principles of Mechanism," was adopted by the Brown & Sharpe 
Company some forty years ago. Since that time this system has come into 
general use. 

THE MOLDING PROCESS 

If a gear blank made of some pliable material is forced into contact with a 
rack, as shown in Fig. 8, the rack tooth would conjugate teeth in the blank. 

It does not matter what form is given the conjugating tooth, as long as it 
has a regular line of action; all gears formed by it will interchange. 

The Bilgram spur and spiral gear generating machine operates upon this 
principle. See Fig. 9. The cutter A, which is a reciprocating or planing tool 
having the profile of a correct rack tooth — namely, a truncated, straight-sided 



TOOTH PARTS 



wedge. While this tool reciprocates, it also travels slowly to the right, the 
blank meanwhile turning under it, the motion being that which would exist 
were the tool a rack tooth and the blank a gear. During this combined move- 




Emery Wheel 



FIG. 9. ACTION OP THE TOOL IN GENERATING A TOOTH. 

ment the tool cuts the tooth space in the manner indicated. In the Bilgram 
bevel-gear machine the tool does not move sidewise, the blank being rolled upon 
it as a complete gear might be rolled on a stationary rack, but in the spur-gear 
machine this action is reversed — the 
blank turning on a fixed center, while 
the tool moves over it, as it would be 
turned by a moving rack. 

The Fellows' gear shaper is designed 
on the same principle, but instead of 
a rack tooth as a planing tool, a gear 
of from 12 to 60 teeth is used, the 
motion of cutter and blank being the 
same as between gears in mesh. See 



Imaginary Rack 



Cutter 








, •> 



FIG. IO. ACTION OF THE FELLOWS' GEAR 
CUTTER. 



Cutter 



FIG. II. GENERATION OF THE FELLOWS ' 
GEAR-CUTTER TEETH. 



Fig. 10. These cutters are ground to shape after being hardened as shown 
in Fig. 11, in which the emery wheel is shaped as the planing tool in Fig. 9. 
The cutter being ground taking the place of the gear. 



IO 



AMERICAN MACHINIST GEAR BOOK 



TO DRAW THE INVOLUTE CURVE 

The involute curve is constructed on the base circle as follows: Draw the 
pitch circle and through pitch point P, Fig. 12, draw the line of action at the 
required angle of obliquity. Tangent to this line draw the base circle. 

Divide the base circle into any number of equal spaces, i', 2' ', 3', 4/, 5', 6', 
as shown in Fig. 13. From each of these points draw lines intersecting at 
center 0. Draw lines i'-i, 2'-2, 3 '-3, etc., tangent to base circle and at right 



angles with lines extending to center. 
one of the divisions of base circle: 
line 2 '-2 equal to two divisions, line 
3 '-3 equal to three divisions, and so 
on. Then through points 1, 2, 3, 4, 



Make the length of line i'-i equal to 





FIG. 12. LOCATING THE BASE CIRCLE. 



FIG. 13. DRAWING THE INVOLUTE. 



5, 6, etc., trace the involute curve. Find a convenient radius, not necessarily 
on base circle, from which to draw the balance of the teeth, several radii 
sometimes being necessary to get the proper curve, especially for a small 
number of teeth. The involute curve does not extend below the base 
circle. 

Below the base circle drawing the teeth is simply a matter of obtaining suf- 
ficient clearance to avoid interference with the teeth of the mating gear. 

SINGLE CURVE TEETH 

This method of drawing gear teeth should be used only when the gear is to 
be pictured, not for templets. It is approximately correct only for 14^ degree 
teeth and for 30 teeth and over, although it. may always be used for the curve 
between the base circle and the pitch circle. 

Referring to Fig. 14, draw the pitch diameter and locate addendum, deden- 
dum, and tooth spaces. With a radius of one half the radius of the pitch circle 
draw semicircle A from the center to the pitch line with the point of dividers 



TOOTH PARTS 



II 



located on the center line midway between these points. Take one half 
of this radius or one quarter the radius of pitch circle and, with point of 



.^ Tooth Curve Radius 

One Quarter of Pitch Radius 




FIG. 14. LAYING OUT A SINGLE CURVE TOOTH. 

dividers at B on pitch circle draw an arc cutting semicircle A at point C. 
This is the center for the first tooth curve and locates the base circle for 
all tooth arcs. 



DEMONSTRATION OF INVOLUTE PRINCIPLE BY A MODEL 

An excellent Reuleaux model for demonstrating the principle of the involute 
system was loaned by Cornell University and is shown in Figs. 15 to 22. 

The segments in the model represent gears of 21 and 17 teeth, about iM 
inch circular pitch. The angle of obliquity is 30 degrees, which is sufficiently 
great to drop the base circle slightly below the bottom of the teeth in the 
smallest gear of the pair, thereby securing a theoretical tooth free from under- 
cut or correction for interference. The teeth are carried to a point to show 
all the tooth action possible. 

The base circles upon which the involute curve is constructed are represented 
in this model by rims E and F, upon which is tightly wrapped the band H, 
which, when wound from the base circle of the gear to the base circle of the 
pinion, represents the line of action, also the angle of obliquity, or pressure, the 






12 



AMERICAN MACHINIST GEAR BOOK 



Tension Spnn 



P*tckbne 




Pitch Line 








1^. £ H 




^ra Jl 


£^i / 




| :S : m A 






Nf B V 






,■■ 


^^^^F 








iBr 








FIGS. 15 AND 17. FIGS. 16 AND 18. 

INVOLUTE GEAR TOOTH MODEL. 



TOOTH PARTS 13 

thrust of the teeth in contact being constantly in the direction of this band, 
which intersects all points of contact between the teeth. 

Referring to Fig. 2, this band represents a line rolled on the base circle, also, 
as is self-evident, a string unwound from its base circle, as in Fig. 3, any given 
point on which will describe the involute curve. 

The points describing teeth in the gear segments are shown on the model by 
the lines a, b, c and d on the band H connecting the base circles, any of which 
will follow the contour of both teeth engaged from top to bottom as the gears 
are rotated, as well as those not yet in action. In fact, the generating of the 
involute curve is begun just as soon as any point on the band is raised from 
the rim representing the base circles and continues until the movement of the 
gear is stopped, the tooth outline being described by one of the points crossing 
the pitch line. The amount of this curve that is used above and below the pitch 
line depends upon the proportionate length of the tooth. 

The location of points a, b, c, d on band H have no significance in the model; 
they are placed to correspond with the location of the teeth, being projected 
on a radial line drawn from the bottom of the tooth curve. If the pitch of 
gear segments had been coarser these points would simply have been farther 
apart. 

In the model, the length of the tooth is restricted only by the meeting 
of the curves describing the opposite sides of the tooth. The tooth is carried 
below the pitch line the same distance as above, plus a sufficient distance for 
clearance. 

In case the gear segments were taken off, the model would simply represent 
the base circles of two gears, connected by a band, the angle of which, from the 
horizontal, would indicate the angle of obliquity. The driven shaft is pro- 
pelled by the band, acting as a belt ; any point upon it will describe the proper 
tooth outline from the base line up, the pitch point being at the intersection 
of this band and the line of centers. See Fig. 4. 

Another important point is shown. The contact point of any two teeth en- 
gaged is followed by one of the points a, b, c, etc., as the gears are rotated and 
the band or line upon which these points are marked is always tangent to the 
two base circles. This illustrates the law of contact for involute teeth 
denned in connection with Fig. 35. The action between two involute teeth is 
that of two cylinders rolling and slipping upon each other. The diameter of 
these cylinders is constantly changing, one becoming larger and the other be- 
coming smaller as the teeth enter and leave contact. The impulse given the 
driven gear will be variable if these conditions are not fulfilled during the en- 
tire action. This illustrates the importance of having the tooth curves theo- 
retically correct. 



14 



AMERICAN MACHINIST GEAR BOOK 














E \^ 






'■-■' P 'r i 




R 




















' * h 








ar* 


vJ5 


£m 


^k 


*^ 


k 2) 




FIGS. 19 AND 21. FIGS. 20 AND 22. 

INVOLUTE GEAR TOOTH MODEL. 



TOOTH PARTS 15 

If the base circles E and F in the model were brought closer together it would 
reduce the pitch circles of both gears proportionately, also reduce the angle of 
obliquity, as the band or line representing the angle of obliquity must always 
be tangent to both base circles and pass through the pitch point, where the 
velocities of gears are equal. Drawing the base circles apart increases the 
pitch circles, also the obliquity, although the action of the teeth remains cor- 
rect as long as they are engaged. See Figs. 20, 21, and 22. 

In Fig. 15, the point b on band H is just touching the point of the tooth A 
as it enters into contact with tooth B. In Fig. 16, it has followed the contact, 
and therefore the outline of both engaging teeth to the pitch point F, and in 
Fig. 17, it is just leaving the point of the tooth B at the end of its contact with 
the tooth A . The point b will continue toward the upper tooth A until it com- 
pletes the involute and comes to rest on the base circle E. 

Figs. 18 and 19 show relation of points c and d with other teeth in the seg- 
ments as the gears are revolving to the left, and affords a better opportunity to 
study the entire action of the model. 

In Figs. 20, 21, and 22 the centers have been widened X A inch. Fig. 20 shows 
tooth A just entering contact with tooth B. In Fig. 21 the point b and band 
has followed the contact to the pitch point, which is now midway between the 
two pitch circles as marked on segments. 

Fig. 22 shows the tooth A leaving contact, giving the entire range of action. 
This illustrates a peculiarity of the involute system, and explains how it is pos- 
sible to obtain correct tooth action if a pair of gears are moved from their 
proper centers. 

It will be noticed that the points a, b, c, and d follow the tooth outlines and 
points of contact just as accurately as when on proper centers, although the 
angle of obliquity is changed. The involute curve is always the same for a 
given base diameter and, as the pitch diameter, and not the base diameter is 
altered, a change in the center distance will make no change in the action of 
the teeth. This is illustrated by Fig. 23. 

There will be new pitch diameters automatically established at the pitch 
point as the centers are moved, simply a different portion of the involute curve 
is used for the tooth. It is apparent that the farther out the tooth is placed 
the greater will be the distance between the pitch and the base circles and the 
greater the angle of obliquity. With the model in this position, if the lines 
drawn on gear segments to represent the pitch diameter were moved out until 
they were rolled together at the pitch point and the teeth made heavier at the 
pitch line to take up backlash, we would have gears of increased obliquity, 
which in turn, could be still farther apart as long as there were any teeth left, 
for with any increase of angle the length of tooth is necessarily shorter. 



i6 



AMERICAN MACHINIST GEAR BOOK 




FIG. 23. DIAGRAM SHOWING HOW PROPER ACTION IS MAINTAINED AS THE GEAR AXES ARE 

SEPARATED. 




TIG. 24. GRAPHICAL DEMONSTRATION FOR INTERFERENCE OF SPUR GEARS. 



TOOTH PARTS 



17 



INTERFERENCE IN INVOLUTE GEARS 

The limitations and inaccuracies of the involute system are well explained in 
the following paragraphs by C. C. Stutz: 

While the general principles governing the interference of involute gears are 
well known, the following graphical demonstrations, formulas, and plotted dia- 
grams may place this general information 
in more efficient form for the use of 
many. 

Fig. 24 shows a graphical demonstra- 
tion of the interference of a 5-pitch, 15- 
tooth true involute form spur pinion and 
a 5-pitch, 48-tooth mating gear. The 
point F is the right-angled intersection 
of a line drawn from the center of the 
pinion, and at an angle of 14M degrees 
with the common center line of the pinion 
and gear, with the line of pressure which 
is drawn through the point of tangency 
of the two pitch circles and at an angle of 
14^ degrees to the common tangent at 
that point. If this point falls within the 
addendum circle of the meshing gear, the 
tooth of the meshing gear will interfere 
from this point up to its addendum circle. 
Therefore the tooth from this point on 
the curve must be corrected to overcome 
it. 

If the point F falls on or outside of the 
addendum circle of the meshing gear no 
interference will result. The point F' for fig. 25. 
an angle of obliquity of 20 degrees falls on 

the addendum circle and thus the gear and pinion indicated in the illustration 
would mesh without interference for this angle. 




INTERFERENCE OE SPUR GEARS. 



FORMULA FOR LOCATING THE POINT OF INTERFERENCE OF SPUR GEARS 

Referring to Fig. 25: 

~LetAF=c. A B = r 2 . A D = d. B E=r 1 . D E=f. 

a = the angle of tooth pressure. 

y= the distance from the center of the gear to the point at which in- 
terference begins. 



l8 AMERICAN MACHINIST GEAR BOOK 

#=the distance from the point at which interference begins to the 
addendum circle of the gear measured along a radius. 

0=the perpendicular distance from the point at which interference 

begins to the center line of the pinion and gear. 

Then 

r 1 = the pitch radius of the gear. 

r. 2 = the pitch radius of the pinion. 

D' = ihe pitch diameter of the gear. 

D =the outside diameter of the gear. 

Then 

c = r 2 cos a, and 
d=c cos a=r 2 cos 2 a. 

Now 

f=r 1 +r 2 -d. 

= r 1 -\-r 2 (i — cos 2 a), 

and 

= c sin a=r 2 sin a cos a. 

Now 

y 2 = f 2 + O 2 and y = \/ p _|_ O 2 . 
Then by substituting 
y = V [r 1 + r 2 (i — cos 2 a)] 2 -f- (r 2 5m a co5 a) 2 
For a pressure angle of 143^ degrees 
y = V ir l + 0.0627 r 2 ) 2 + (0.2424 rj 2 , 
and 

a; = y. 

2 

For a pressure angle of 20 degrees 

y = V (r x + 0.1169 r 2 ) 2 + (0.3214 r 2 ) 2 , 

and 

D 

x = y. 

2 

Solving for x and y will give the point of interference for any particular case. 

DIAGRAM FOR LOCATION OF INTERFERENCE 

Fig. 26 shows a diagram giving the location of the point of the beginning of 
interference for one diametral pitch involute gears from 10 to 135 teeth mesh- 
ing with a 12-tooth pinion. The ordinates are the distances from the point 
where interference commences to the addendum circle of the gear measured 
along the radius. They correspond to the quantity x in the preceding equa- 



TOOTH PARTS 



19 



















































































































































































1 




- 
































































s 




















s 

3 








£ 






















S 




























T3 
S3 

T3- 








•? 

C 

0) 

m 1 






















3 








i 




















«! 








(© 1 

1 






















•73 








(3 

3 




















,0 

4-> 

7) 








3 

|d 1 






















35 

3 








.2 




























































t 




















o> 1 










"bO" 

0> 






















1 

0) 




























i 










en 

CO 

0) 






















1 

to 








en 

0) 




















01 










m 

0) 






















en 

0) 








0) 






















bO" 
^r 4 










p 






















bo 
l<u 

P 






i-t 
1 


1 


















■£ 

I 
































1 








W 
> 




















> 










1 

1 




















P 

> 








g 




















PA 
P 
O 










si 




















p 


1 
1 
























































































1 



























































































































































































































































































































































































































































































































































































































































to 

o> . 

J-l 

8 3 

(1) 



H 
o 

B 



o o 



o o 



_ OOOOOOOOOOOOOOOOOOOOOOOO 

»*NOOOt0^cqOCCtO<*NOCO(O^NOOO<OrtiNOCC(D^(NoC<ltOT|iN 
O ffl O IO. U U 111 IO ^ ^ ^ ^ ^ W M M M M CI N N N W H H H H h O O O O 
OOOOOOOOOOOOOOOOOOOOOOO'OOOOOOOOO 



Interference in Inches 



20 



AMERICAN MACHINIST GEAR EOOK 



tions. From this point to the addendum circle the tooth outline must be cor- 
rected. 

The upper curve A is for a pressure angle of 143^ degrees and an addendum of 
0.3183 X circular pitch. The second, B, is for the same pressure angle and a 
shorter addendum, 0.25 X circular pitch. 

This addendum factor is for what is known as the stubbed tooth standard, 
as proposed by the author on page 23. 

The third curve, C, is for a pressure angle of 20 degrees and an addendum of 
0.3183 X circular pitch, while the lowest one, D, is for the 20-degree angle and 
the stubbed tooth addendum. 

The diagram as plotted is for one diametral pitch. To find the correspond- 
ing ordinate for any other pitch divide the value given in the diagram by the 
required pitch. The quotient will be the distance desired. 

INTERFERENCE OF RACK AND PINION 

Interference will occur between the teeth of a rack and pinion when the point 
B, Fig. 27, which is the intersection of a perpendicular from the point O to the 
line of pressure A L falls inside of the rack addendum line E E. In the figure 




FIG. 27. INTERFERENCE OF GEAR AND RACK. 



the distance over which interference takes place is C D. It is usual practice to 
shorten the rack teeth by the amount of this interference and the following 
equations give an easy method of computing this distance. 



Let 



Let 



TOOTH PARTS 21 



TV = the number of teeth in the pinion. 

p=ihe diametral pitch. 

r=the pitch radius. 

& = the radius of the base circle. 

a=the pressure angle. 

#=the distance necessary to shorten the addendum of the rack tooth 

and 

5= the normal addendum of the rack tooth. 



Then 



^ = 



r = 



i 



b = r cos a, 
D = b cos a, 
D = r cos 2 a } 
OC =r- s, 

x =0 D — C, and substituting 
= r cos 2 a — (r — s) 

HN i 

(cos 2 a — i) -f- 



Whence 



P P 

I -- V 2 N (i - COS 2 a) 



X = . 

p 



For a pressure angle of 14^ degrees 

1 - 0.03135 TV 



x = 



P 
For a pressure angle of 20 degrees 

1 — 0.05849 TV 



x = 



Solving these equations we find that for the true involute form of tooth and 
a pressure angle of 143^2 degrees interference between the teeth of rack and pin- 
ion begins with a pinion of 31 teeth. Similarly for a 20-degree pressure angle 
the interference begins with a pinion of 1 7 teeth. 



22 



AMERICAN MACHINIST GEAR BOOK 



INTERFERENCE OF INTERNAL GEAR AND PINION 






The following method of correction and equations are true for all combina- 
tions when the pinion has less than 55 teeth. 
Referring to Fig. 28: 




Let 



FIG. 28. INTERFERENCE OF INTERNAL GEAR AND PINION. 



r, = the pitch radius of gear. 
r, 2 = the pitch radius of pinion. 
b = the radius of the base circle. 
c = the radius of the correction circle. 
d = the radius of the rounding off circle. 
e = the radius of the interference circle. 
= the radius of the tooth cutting. 
a = the pressure angle. 



Then 



b = r x cos a, 
e = H (b + rX 

COS a, 

= \i r, , and 



TOOTH PARTS 



23 



EXISTING TOOTH STANDARDS— BROWN & SHARPE'S* 

The Brown & Sharpe system is perhaps the best known; the angle of 
obliquity being 14^ degrees. 

Addendum, = 0.3183 p 1 or 

1 1 ^7 
Dedendum, = 0.3683 p 1 or 

2 
Working depth, = 0.6366 p 1 or — - 

Whole depth, = 0.6866 p l or ^ I ^- 

™ 1 0.11:7 
Clearance, = 0.05 p 1 or ±L ~ 

In which p 1 = circular pitch, and p = diametral pitch. 

GRANT'S 

The Grant system has an angle of obliquity of 15 degrees, otherwise it is the 
same as Brown & Sharpe's. This system is used on the Bilgram generator. 

SELLERS' 

Wm. Sellers & Co. adopted a form of tooth some 32 years ago in which 
the angle of obliquity was 20 degrees, with an addendum of 0.3 and a clear- 
ance of 0.05 of the circular pitch. 

HUNT'S 

The C. W. Hunt Co. have a standard in which the angle of obliquity is 
14^ degrees; the tooth parts being as follows: 



Addendum, = 0.25 p\ or 
Dedendum, = 0.30 p 1 , or 


P 
0.9424 

P 


Working depth, = 0.50 p 1 , or 


1.5708 
P 


Whole depth, = 0.55 p\ or 
Clearance, = 0.05 p\ov 


1.7278 

P 
O.I57 

P 


THE AUTHOR'S 





This system, presented in connection with a discussion of an interchangeable 
involute gear-tooth system at the December, 1908, meeting of the American 
Society of Mechanical Engineers, was originally published in American 



24 



AMERICAN MACHINIST GEAR BOOK 



Machinist, June 6, 1907. Angle of obliquity is 20 degrees. Balance of the 
tooth parts being the same as the Hunt system described above. 

FELLOWS' 

The stubbed tooth adopted by the Fellows Gear Shaper Company has an 
angle of obliquity of 20 degrees. The tooth parts, however, do not bear a 
definite relation to the pitch; the addendum being made to correspond to a 
diametral pitch one or two sizes finer, as: 

Actual pitch _ 2 2 Y A 3 4 5 

* 5 7 



10 



10 
12 



12 



18 



Pitch depth 2^2 3 

The upper figures indicate the diametral pitch for tooth spacing and the 
lower figures indicate the diametral pitch from which the depth is taken. 
In this system the addendum varies from 0.264 to 0.226 of the circular pitch; 




Fig. 30 

Proposed Stubbed Involute Tooth Shape 

12 Teeth, 20 Degrees. 

Addendum=£L25 X Circular Pitch. 

COMPARATIVE FORMS OF 14^-DEGREE AND 20-DEGREE STANDARDS. 

0.25, which is the addendum for the Hunt and the author's standard, is a 
rough mean. 

The author's standard tooth is shown in Fig. 30 for comparison with the 
14^-degree standard in Fig. 29. 

Wilfred Lewis discussed tooth standards before the American Society of 
Mechanical Engineers, 1900, as follows: 

"About 30 years ago, when I first began to study the subject, the only system 



TOOTH PARTS 25 

of gearing that stood in much favor with machine-tool builders was the cy- 
cloidal. 

" For some time thereafter William Sellers & Co., with whom I was connected, 
continued to use cylindrical gearing made by cutters of the true cycloidal shape, 
but the well-known objection to this form of tooth began to be felt, and pos- 
sibly 25 years ago my attention was turned to the advantages of an involute 
system. The involute systems in use at that time were the ones here de- 
scribed as standard, having 14^ degrees' obliquity, and another recommended 
by Willis having an obliquity of 15 degrees. Neither of these satisfied the re- 
quirements of an interchangeable system, and with some hesitation I recom- 
mended a 20-degree system, which was adopted by William Sellers & Co., and 
has worked to their satisfaction ever since. I did not at that time have quite 
the courage of my convictions that the obliquity should be 223^ degrees or 
one-fourth of a right angle. Possibly, however, the obliquity of 20 degrees 
may still be justified by reducing the addendum from a value of one to some 
fraction thereof, but I would not undertake at this time to say which of the 
two methods I would prefer. 

"I brought up the same question nine years ago before the Engineers' Club 
of Philadelphia, and said at that time that a committee ought to be appointed 
to investigate and report on an interchangeable system of gearing. We have 
an interchangeable system of screw threads, of which everybody knows the 
advantage, and there is no reason why we should not have a standard system 
of gearing, so that any gear of a given pitch will run with any other gear of the 
same pitch. " 

A UNIVERSAL STANDARD 

To do away with the multiplicity of standards, especially in connection 
with the involute form of tooth, and bring a universally accepted standard 
gear-tooth system out of the present chaos, a special committee was appointed 
by the American Society of Mechanical Engineers to recommend a standard 
for involute, interchangeable gear teeth. This committee, which was com- 
posed of five members under the chairmanship of Wilfred Lewis, accomplished 
little and after several years of deliberation has finally given up the work. 
Notwithstanding the failure of the committee to render any unanimous 
recommendation or to accomplish anything of importance, hope has not 
been lost of eventually bringing about this much-needed reform. A second 
committee will probably be appointed by the Society to continue the work. 

While the committee was active many circular letters were sent out to 
various gear manufacturers requesting their individual opinions and their 
co-operation in arriving at a standard type of tooth. Hundreds of replies 



26 



AMERICAN MACHINIST GEAR BOOK 



were received by the committee and much interest was expressed in the 
work, but the preferences and conclusions sent in were extremely conflicting. 
Experiments to ascertain the friction losses of involute teeth of different 
obliquity were conducted at certain technical laboratories, but the results 
obtained were neither convincing nor entirely satisfactory. 

If the investigations of the committee showed anything, it was doubt as 
to whether the most commonly employed standard of 14^ degrees involute 
was really the best suited for a universal standard. 



MODIFIED TEETH 

A common method of modifying the involute tooth to avoid either inter- 
ference, undercutting, or the necessity of departing from the true outline is by 
shortening the dedendum and lengthening the addendum of the pinion tooth. 
The opposite treatment is given the gear tooth, the dedendum being made 
deeper to accommodate the added addendum of the pinion and the addendum 
of the gear correspondingly short. This method is employed on all bevel-gear 
generating machines for angles less than 20 degrees to avoid interference, the 
amount of correction depending, of course, upon both the number of teeth being 

cut and the number of teeth in the engag- 
ing gear, or, in other words, depending 
upon the position of the base line. 

On bevel-gear generating machines it is 
the practice to make no modification in 
the angle for a 20-degree tooth when cut- 
ting a depth equal to o.6866p f . For this 
depth of tooth and a pressure angle of 20 
degrees interference beginning at 17 teeth, 
enough roll, however, can be given the 
blank to allow the generating tool to un- 
dercut the flank of the tooth, and avoid interference without any correction 
of the tooth parts. This is not the case, however, when cutting the standard 
14^ or 15 degrees on account of limitation in the movements of the machine. 
This modification in the tooth parts for bevel gears is accomplished by shift- 
ing the face angles and outside diameters, the pinion being enlarged and the 
gear reduced. 

The dedendum of the pinion is sometimes shortened for another reason: 
Often the bore is so large as to leave insufficient stock between the bottom of 
the teeth and the keyseat. See Fig. 31. When the pinion cannot be enlarged 
or the bore reduced the only possible recourse is to shorten the dedendum, 
taking the amount shortened from the point of the gear tooth. This practice 




FIG. 



12 Teeth 

3 Diamjetral Pitch 

2H Inch Bore 



31. SHORTENING THE DEDENDUM 
TO STRENGTHEN KEYWAY. 



TOOTH PARTS 



27 



is not to be recommended although extensively used; it would be much better 
to apply the short tooth of increased obliquity to such cases. 

THE OCTOID 

All bevel-gear generating machines operate on the octoid system, and not the 
involute, as is generally supposed. 

An involute spur gear may be generated by the action of a tool representing 
the rack tooth, as illustrated by Fig. 9. In generating a bevel gear, however, 
the tool representing the engaging rack tooth must always travel toward the 
apex of the gear being cut, swinging in a partial circle instead of travelling on 



7°t 

Involute Tooth 





Octoid Tooth 
Fig. 33. 



FIG. 32. GENERATING THE OCTOID TOOTH. 



a straight line in the direction of the rotation of the gear, as is the case when 
cutting a spur gear. The base of the bevel-gear tooth is, therefore, a crown gear 
instead of a rack. 

An involute crown gear theoretically correct will have curved instead of 
straight sides as shown in Fig. 32. As it is not practical to make the generating 
tools this peculiar shape, they are made straight sided and the octoid tooth is 
the result. 

THE LINE OF ACTION 

There is a definite relation between the circle or line which will describe the 
tooth outline and the line of action. Thus, if the line of action is in the form of 
a circle, as shown in Fig. 34, that circle of which this line is a segment will de- 
scribe the tooth outline if rolled upon the pitch circle. The difficulties encoun- 



28 



AMERICAN MACHINIST GEAR BOOK 



tered in the general application of this law are well illustrated by George B. 
Grant in section 32 of his "Treatise on Gear Wheels," as follows: "This ac- 
cidental and occasionally useful feature of the rolled curve has generally been 
made to serve as a basis for the general theory of the tooth curve, and it is re- 
sponsible for the usually clumsy and limited treatment of that theory. The 
general law is simple enough to define, but it is so difficult to apply, that but 
one tooth curve, the cycloidal, which happens to have the circle for a roller, 

can be intelligently handled by it, and 
the natural result is, that that curve has 
received the bulk of the attention. 

For example, the simplest and best of 
all the odontoids (pure form of tooth 
curve), the involute, is entirely beyond 
its reach, because its roller is the loga- 
rithmic spiral, a transcendental curve 
that can be reached only by the higher 
mathematics. 

No tooth curve, which, like the in- 
volute, crosses the pitch line at any 
angle but a right angle, can be traced 
by a point in a simple curve. The trac- 
ing point must be the pole of a spiral, and therefore a mechanical impossibility. 
A practicable rolled odontoid must cross the pitch line at right angles. 

To use the rolled curve theory as a base of operations will confine the dis- 
cussion to the cycloidal tooth, for the involute can only be reached by abandon- 
ing its true logarithmic roller, and taking advantage of one of its peculiar prop- 
erties, and the segmental, sinusoidal, parabolic, and pin tooth, as well as most 
other available odontoids, cannot be discussed at all. " 




FIG. 34. RELATION OF THE LINE OF 
ACTION TO THE DESCRIBING CIRCLE. 



THE LAW OF TOOTH CONTACT 

To transmit uniform motion, any form of tooth curve is subject to this gener- 
al law: "The common normal to the tooth must pass through the pitch point. " 
That is, a line drawn from the pitch point P through the contact point of any 
pair of teeth, as at b, must be at right angles or normal to the engaging portions 
of both teeth. See Fig. 35. 

In the involute system the line of action always passes through the pitch 
point P, and the engaging teeth take their base from the points / and y, 
where the line of action intersects the base circles. Conversely, a line drawn 
from the instantaneous radii of any two teeth engaged will pass through their 



TOOTH PARTS 



29 



point of contact if the teeth are correctly formed. For example: In Fig. 35, the 
point of contact between the teeth C and D is at b, on the line of action, the 




FIG. 35. THE ARC OF ACTION. 



radius of the engaged portion of the tooth C is at /, and the radius of the tooth 
D is at y, fulfilling the required conditions. 



THE ARC OF ACTION 

The tooth action between two gears is between the points a and b, at which 
points the line of action intersects the addendum circles of the two gears. The 
actual length of contact is along the pitch lines occupied by the teeth whose 
addendum circles intersect the line of action, or between the points c and d. 
See Fig. 35. 

The distance P — d passed over while the point of contact approaches the 
pitch point is the arc of approach, the distance P — c passed over as the point 
of contact leaves the pitch point is the arc of recess. 

By increasing the addendum of the driving gear the arc of approach is re- 
duced and the arc of recess is increased. The friction of the arc of approach is 
greater than in the arc of recess. 



3o 



AMERICAN MACHINIST GEAR BOOK 



THE BUTTRESSED TOOTH 

The buttressed tooth shown in Fig. 36 is described by Professor Robert 
Willis in a paper published in the Transactions of the Institute of Civil Engi- 
neers, London, 1838. It is apparent that the object is to obtain a strong tooth 




11 Teeth 
/ 2V 2 Inch Pitch 



FIG. 36. THE BUTTRESSED TOOTH. 

for a pair of gears operating continuously in one direction. This is accomplished 
by increasing the angle of obliquity of the back of the tooth, the face of the 
tooth being any angle desired. If the back of the tooth is correctly formed the 




PIG. 37. BUTTRESSED TEETH IN CONTACT 



gears will operate satisfactorily in either direction although with an increased 
pressure on their bearings when using the back face of the teeth owing to the 



TOOTH PARTS 3 1 

increased obliquity of action. For many purposes there is no objection to this, 
and it is a great wonder that this tooth is not more extensively used. 

Of course, there must be a limit to the angle of the back of the tooth. For 
practical purposes the curve at the top of tooth at the back should not extend 

further than the center line of the tooth; for an addendum of — or 0.6866^,, 

P 
this will occur at an angle of about 32 degrees. A greater angle than this will 
subject the tooth to breakage at the point. In Fig. 37 is shown a pair of but- 
tressed tooth gears in contact. 

STEPPED GEARS 

A stepped spur gear consists of two or more gears keyed to the same shaft, 
the teeth on each gear being slightly advanced. If mated with a similar gear 
the tooth contact will be increased, which increases the smoothness of action. 
A common form of this type of gear is that of two gears cast in one piece with 
a separating shroud. For a cut gear there must be a groove turned between 
the faces of sufficient width to allow the planing tool or cutter to clear. A 
tooth is placed opposite a space, when the gear is made in two sections. 

HUNTING TOOTH 

It has been customary to make a pair of cast tooth gears with a hunting 
tooth, in order that each tooth would engage all of the teeth in the mating gear, 
the idea being that they would eventually be worn into some indefinite but 
true shape. Some designers have even gone so far as to specify a pair of " hunt- 
ing-tooth miter gears." That is, one " miter" gear would have, say, 24 teeth 
and its mate 25 teeth. 

There never was any call for the introduction of the hunting tooth even in 
cast gears, but in properly cut gears any excuse for its use has certainly ceased 
to exist. 

TEMPLET MAKING 

In making the templets for gear teeth there are several points of importance 
to be kept in mind, namely: 

Templets should be made of light sheet steel instead of zinc which is often 
employed; the surface of steel should be coppered by the application of blue 
vitriol. 

For spiral or worm gears, templets should always be made for the normal 
pitch. 

For spacing and tooth thickness, always use chordal measurements. Check 
the chordal distance over the end teeth of templet. This is of the utmost im- 
portance. 



32 



AMERICAN MACHINIST GEAR BOOK 



Put enough teeth in the templet to show the entire tooth action, and try the 
templets on centers before making up the cutters or formers. 

It is a good idea to make a standard 
templet of each pitch as they are required, 
to try out other templets that must be made 
later on. 

When a templet is required for a fine pitch 
gear it is good practice to lay out the teeth 
on white paper 10 or even 20 times the actual 
size and reduce by photography. On this 
drawing the center should be plainly marked 
and inclosed in a heavy circle, also a short 
section of the pitch line should be made 
heavy with a connecting radial line indicat- 
ing the radius of pitch circle. 

If the pitch radius required is 1^ inches, 
it should be made, say, 15 inches on the drawing. The drawing is then photo- 
graphed, the camera being set until the radial line, which was drawn 15 inches, 
measures i3^ inches on the ground glass. See Fig. 38. 

DEFINITION OF PITCHES 

Diametral pitch is the number of teeth to each inch of the pitch diameter. 
Circular pitch is the distance from the edge of one tooth to the corresponding 
edge of the next tooth measured along the pitch circle. 




FIG. 38. TOOTH OUTLINE AS PHOTO- 
GRAPHED FROM LARGE SCALE 
DRAWING. 



Addendum 




TOOTH PARTS. 



TOOTH PARTS 



33 



DIAMETRAL 
PITCH 


CIRCULAR 
PITCH 


THICKNESS OF 

TOOTH OF 

PITCH LINE 


WHOLE DEPTH 


DEDENDUM 


ADDENDUM 


H 


6.2832" 


3.I416" 


4-3I42" 


2.3142" 


2.0000" 


% 


4.1888 


2.O944 


2.8761 


I.5728 


*-3333 


i 


3.1416 


I.5708 


2.1571 


I-I57I 


I .OOOO 


iH 


2-5I33 


I.2566 


I-7257 


O.9257 


O.80OO 


iVi 


2.O944 


I.O472 


I.4381 


O.7714 


O.6666 


*H 


1-7952 


O.8976 


I.2326 


O.6612 


0.57I4 


2 


1.5708 


O.7854 


I.0785 


0.5785 


O.50OO 


^A 


1-3963 


O.6981 


O.9587 


0.5I43 


• O.4444 


^A 


1.2566 


O.6283 


O.8628 


O.4628 


O.4OOO 


2% 


1. 1424 


O.5712 


O.7844 


O.4208 


O.3636 


3 


1.0472 


O.5236 


O.719O 


0.3857 


0.3333 


iVi 


0.8976 


O.4488 


O.6163 


0.3306 


O.2857 


4 


0.7854 


O.3927 


0-5393 


O.2893 


O.250O 


5 


0.6283 


O.3142 


O.4314 


O.2314 


0.2000 


6 


0.5236 


O.2618 


0.3595 


O.I928 


O.1666 


7 


0.4488 


O.2244 


O.3081 


O.1653 


O.1429 


o 
O 


©.3927 


O.I963 


O.2696 


O.I446 


O.I25O 


9 


0.3491 


O.I745 


O.2397 


O.I286 


O.IIII 


IO 


0.3142 


O.1571 


0.2I57 


O.II57 


O.IOOO 


ii 


0.2856 


O.1428 


O.1961 


O.IO52 


0.0909 


12 


0.2618 


O.I309 


O.1798 


O.O964 


0.0833 


13 


0.2417 


O.I208 


O.1659 


O.089O 


0.0769 


14 


0.2244 


O.II22 


O.1541 


O.0826 


0.0714 


IS 


0.2094 


O.IO47 


O.1438 


O.0771 


0.0666 


16 


0.1963 


O.O982 


O.I348 


O.0723 


0.0625 


17 


0.1848 


O.O924 


O.I269 


O.0681 


0.0588 


18 


O.I754 


O.0873 


O.II98 


O.0643 


0.0555 


19 


0.1653 


O.0827 


O.H35 


O.0609 


0.0526 


20 


0.1571 


O.O785 


O.IO79 


O.0579 


0.0500 


22 


0.1428 


O.O714 


O.O980 


O.0526 


0.0455 


24 


0.1309 


O.0654 


O.0898 


O.0482 


0.0417 


26 


0.1208 


O.0604 


O.0829 


O.0445 


0.0385 


28 


0.1122 


O.O561 


O.0770 


O.O413 


0.0357 


30 


0.1047 


O.O524 


O.0719 


O.O386 


0.0333 


32 


0.0982 


O.O49I 


O.0674 


O.O362 


0.0312 


34 


0.0924 


O.O462 


O.0634 


O.O34O 


0.0294 


36 


0.0873 


O.O436 


O.0599 


O.O32I 


0.0278 


38 


0.0827 


O.O413 


O.0568 


O.O304 


0.0263 


40 


0.0785 


O.O393 


O.0539 


O.O289 


0.0250 


42 


0.0748 


O.O374 


O.0514 


O.0275 


0.0238 


44 


0.0714 


0.0357 


O.O49O 


O.O263 


0.0227 


46 


0.0683 


O.O34I 


O.O469 


O.O252 


0.0217 


48 


0.0654 


O.O327 


O.O449 


O.024I 


0.0208 


5o 


0.0628 


O.O314 


O.043I 


O.O23I 


0.0200 


56 


0.0561 


O.O280 


0.9385 


0.0207 


0.0178 


60 


0.0524 


O.O262 


O.O360 


O.OI93 


0.0166 



Table i — Diametral Pitch 
Relation between Diametral Pitch and Circular Pitch, with corresponding Tooth Dimensions 



34 



AMERICAN MACHINIST GEAR BOOK 



CIRCULAR 
PITCH 


DIAMETRAL 
PITCH 


THICKNESS OF 

TOOTH OF 

PITCH LINE 


WHOLE DEPTH 


DEDENDUM 


ADDENDUM 


6 " 


O.5236 


3. OOOO" 


4.II96" 


2.2098" 


1. 9098" 


5 " 


O.6283 


2.50OO 


34330 


I.8415 


I-59I5 


4 " 


0.7854 


2.0000 


2.7464 


1.4732 


I.2732 


3^" 


O.8976 


I.75OO 


2.403I 


I.289O 


I.II40 


3 " 


I.O472 


I.5OOO 


2.0598 


I. IO49 


0-9550 


2%" 


1. 1424 


I-3750 


1.8882 


I.OO28 


O.8754 


2^" 


I.2566 


I.250O 


1. 7165 


O.9207 


O.7958 


2^" 


I-3963 


1. 1250 


1-5449 


O.8287 


O.7162 


2 " 


I.5708 


I. OOOO 


1-3732 


0.7366 


O.6366 


J.%" 


1-6755 


0-9375 


1.2874 


O.6906 


0.596ft 


1%" 


1-7952 


O.8750 


1. 2016 


O.6445 


0.5570 


iVs" 


1-9333 


O.8125 


1.1158 


0.5985 


0.5I73 


lV2" 


2.0944 


0.7500 


1.0299 


0.5525 


0.4775 


ifc" 


2-1855 


O.7187 


0.9870 


O.5294 


O.4576 


iVs" 


2.2848 


O.6875 


0.9441 


O.5064 


0.4377 


*%" 


2.3936 


O.6562 


0.9012 


0.4837 


O.4178 


iH" 


2.5133 


O.6250 


0.8583 


O.4604 


O.3979 


i 3 A" 


2.6465 


0.5937 


0.8156 


0.4374 


0.3780 


i.Vs" 


2.7925 


O.5625 


0.7724 


O.4143 


O.3581 


i^e" 


2.9568 


0.53*2 


0.7295 


0.39I3 


O.3382 


i " 


3.1416 


O.50OO 


0.6866 


O.3683 


O.3183 


%" 


3-35io 


O.4687 


0.6437 


0-3453 


O.2984 


Vs" 


3-5904 


0-4375 


0.6007 


O.3223 


O.2785 


%" 


3.8666 


O.4062 


0-5579 


O.2993 


O.2586 


%" 


4.1888 


0.3750 


0.5150 


O.2762 


O.2387 


%" 


4.5696 


0.3437 


0.4720 


O.2532 


O.2189 


Vs" 


5.0265 


0.3I25 


0.4291 


O.230I 


O.1989 


%" 


5-585I 


O.2812 


0.3862 


O.2071 


O.179O 


V*" 


6.2832 


O.250O 


o.3433 


O.1842 


O.1592 


%" 


7.1808 


O.2187 


0.3003 


O.161I 


O.I393 


2 // 
5 


7.8540 


0.2000 


0.2746 


O.I473 


O.1273 


Vs" 


8.3776 


O.1875 


0.2575 


O.I381 


O.II94 


Vs" 


9.4248 


O.1666 


0.2287 


O.I228 


O.I061 


%r 


10.0531 


O.1562 


0.2146 


O.IISI 


O.0995 


2 // 
7 


10.9956 


O.I429 


0.1962 


O.IO52 


O.O909 


Vi" 


12.5664 


O.I250 


0.1716 


O.O92I 


O.0796 


2 /' 
IT 


14.1372 


O.IIII 


0.1526 


O.0818 


O.0707 


i" 


15.7080 


O.IOOO 


o.i373 


O.0737 


O.0637 


^e" 


16.7552 


0.0937 


0.1287 


O.069O 


O.0592 


^" 


18.8496 


0.0833 


0.1144 


O.0614 


O.053I 


7 


21.9911 


0.0714 


0.0981 


O.O526 


O.0455 


1/" 


25.1327 


0.0625 


0.0858 


O.O460 


O.0398 


i // 

9 


28.2743 


0.0555 


0.0763 


O.O409 


O.0354 


1 // 
1 


3I-4I59 


0.0500 


0.0687 


O.O368 


O.0318 


X*" 


50.2655 


0.0312 


0.0429 


O.O23O 


O.OI99 



Table 2 — Circular Pitch 
Relation between Circular Pitch and Diametral Pitch, with corresponding Tooth Dimensions 



TOOTH PARTS 



35 



JWI 

20 D. P. 
0.1571 Inch C. P. 



18 D. P. 
0.1745 Inch C. P. 



AAA 

16 D. P. 
0.1963 Inch C. P. 



AAA 

14 D. P. 
0.2244 Inch C. P. 



12 D. P. 

0.2618 Inch C. P. 




10 D. P. 
0.3142 Inch C. P. 




9 D. P. 
0.3491 Inch C. P. 




8 D. P. 
0.3927 Inch C. P. 




7 D. P. 
0.4488 Inch C. P, 




6 D. P. 
0.5236 Inch C. P. 




5 D. P. 
0.6283 Inch C. P. 




4 D.P. 
0.7854 Inch C. P. 




3 D.P. 
1.0472 Inch C. P. 



COMPARATIVE SIZES OE GEAR TEETH — INVOLUTE FORM. 



3^ 



AMERICAN MACHINIST GEAR BOOK 




2V 2 D.P. 
1.2566 In. C.P. 




2 D.P. 
1.5708 In. C.P. 




1 3 4 D.P. 
1.7952 In. C.P. 




m D.P. 
2.0944 In. C.P. 

COMPARATIVE SIZES OF GEAR TEETH — INVOLUTE FORMS. 



V 



TOOTH PARTS 



37 




m d. p. 

2.5133 Inch C. P. 




1 D.P. 
3.1416 Inch C. P. 

COMPARATIVE SIZES OF GEAR TEETH — INVOLUTE FORM. 



38 



AMERICAN MACHINIST GEAR BOOK 



NO. 


PITCH 


NO. 


PITCH 


NO. 


PITCH 


NO. 


PITCH 


TEETH 


DIAMETER 


TEETH 


DIAMETER 


TEETH 


DIAMETER 


TEETH 


DIAMETER 


8 


2.550 


43 


13-687 


78 


24.828 


113 


35-968 


9 


2.8/0 


44 


I4.O06 


79 


25.146 


114 


36.286 


IO 


3-l8 3 


45 


14.324 


80 


25-465 


115 


36.605 


II 


3-50I 


46 


14.642 


81 


25-783 


Il6 


36.923 


12 


3.820 


47 


14.961 


82 


26.IOI 


117 


37.24I 


13 


4.I38 


48 


15-279 


83 


26.420 


Il8 


37o6o 


14 


4-456 


49 


15-597 


84 


26.738 


119 


37.878 


15 


4-775 


50 


I5-9I5 


85 


27.056 


I20 


38.196 


16 


5-093 


5i 


16.234 


86 


27-375 


121 


38.514 


17 


5-4" 


52 


16.552 


87 


27.693 


122 


38.833 


18 


5-730 


53 


16.870 


88 


28.011 


123 


39.I5I 


19 


6.048 


54 


17.189 


89 


28.330 


124 


39.469 


20 


6.366 


55 


17-507 


90 


28.648 


125 


39788 


21 


6.684 


56 


17.825 


91 


28.966 


126 


4O.I06 


22 


7.003 


57 


18.144 


92 


29.284 


127 


40.424 


23 


7-321 


58 


18.462 


93 


29.603 


128 


40.743 


24 


7-639 


59 


18.780 


94 


29.921 


129 


41.061 


25 


7.958 


60 


19.099 


95 


30.239 


I30 


41-379 


26 


8.276 


61 


19.417 


96 


30.558 


131 


41.697 


27 


8-594 


62 


19-735 


97 . 


30.876 


132 


42.O16 


28 


8.913 


63 


20.053 


98 


3i-i94 


133 


42.334 


29 


9.231 


64 


20.372 


99 


3I-5I3 


134 


42.652 


30 


9-549 


65 


20.690 


100 


31-831 


135 


42.971 


31 


9.868 


66 


21.008 


101 


32.148 


I36 


43.289 


32 


10.186 


67 


21.327 


102 


32.468 


137 


43.607 


33 


10.504 


68 


21.645 


103 


32-785 


138 


43.926 


34 


10.822 


69 


21.963 


104 


33-103 


139 


44-243 


35 


11. 141 


70 


22.282 


105 


33-421 


I40 


44.562 


36 


n-459 


7i 


22.600 


106 


33-74Q 


141 


44.881 


37 


11.777 


72 


22.918 


107 


34-058 


I42 


45-199 


38 


12.096 


73 


23- 2 37 


108 


34-376 


143 


45-517 


39 


12.414 


74 


2 3-555 


109 


34.695 


144 


45.835 


40 


12.732 


75 


23-873 


no 


35-OI3 


145 


46.154 


4i 


13-051 


76 


24.192 


III 


35-331 


I46 


46.472 


42 


13-369 


77 


24.510 


112 


35-650 


147 


46.79O 



Table 3 — Pitch Diameters for One-Inch Circular Pitch 
Teeth from 8 to 147 

EOR ANY OTHER PITCH MULTIPLY BY THAT PITCH 



METRIC PITCH 



The module is the addendum, or the pitch diameter in millimeters divided 
by the number of teeth in the gear. 



TOOTH PARTS 



39 



The pitch diameter in millimeters is the module multiplied by the number 
of teeth in the gear. All calculations are in millimeters. 
M = module (addendum) 
D' = pitch diameter 
D = outside diameter 
N = number of teeth 
W = working depth of tooth 
W' = whole depth of tooth 
/ = thickness of tooth at pitch line 
/ = clearness 
r = root 
D 



M = 



N = 



N 
D' 



or 



or 



M 

t = M 1.5708, 



N + 2' 

D_ 

M 



I 



D f -- ■- NM, D = (N + 2) M 

W' = W + / 
or M 0.157, r = M + f 



W = 2M 

_M 1.5708 



10 





ENGLISH 




ENGLISH 




ENGLISH 


MODULE 


DIAMETRAL 


MODULE 


DIAMETRAL 


MODULE 


DIAMETRAL 




PITCH 




PITCH 




PITCH 


0.5 


50.800 






7 


3.628 


I 


25.40O 


3 


8.466 


8 


3-175 


1. 25 


2O.32O 


3-5 


7-257 


9 


2.822 


i-5 


16.933 


4 


6.350 


10 


2.540 


I.7S 


I4-5I4 


4-5 


5-644 


11 


2.309 


2 


I2.7OO 


5 


5.080 


12 


2. II7 


2.25 


11.288 


5-5 


4.618 


14 


I.814 


2-5 


IO.160 


6 


4-233 


16 


1.587 


2.75 


9.236 











Module in Millimeters 
Table 4 — Pitches Commonly Used 

CHORDAL PITCH 

The chordal pitch is the shortest distance between two teeth measured on the 
pitch line, in other words, the distance to which the dividers would be set to 
space around the gear on the pitch line. This pitch is not used except for laying 
out large gears and segments that cannot be cut on a gear cutter. For such 
cases, also for laying out templets, it is absolutely necessary to use the chordal 
pitch, as the chordal pitch of the pinion is different from the chordal pitch of 
the gear, the circular pitch of each being equal. 

N = number of teeth, 

C' = chordal pitch, 



40 



AMERICAN MACHINIST GEAR BOOK 



D r = pitch diameter, 
e = sine of one half of angle subtended by side at center. 

180 



e = sine 



N 



D' = 



C 



C' = D' e, or D f sine 



i8o c 
N 



Table 5, diameters for chordal pitch, will be found useful for sprocket gears. 



NO. 


PITCH 


xo. 


PITCH 


xo. 


PITCH 


XO. 


PITCH 


TEETH 


DIAMETER 


TEETH 


DIAMETER 


TEETH 


DIAMETER 


TEETH 


DIAMETER 


4 


I-4I4 


39 


12.427 


74 


23.562 


109 


34-70I 


5 


1. 701 


40 


I2.746 


75 


23.880 


no 


35-OI9 


6 


2.000 


41 


I3.064 


76 


24.198 


III 


35-337 


7 


2-305 


42 


I3-382 


77 


24-5I7 


112 


35.655 


8 


2.613 


43 


I3.699 


78 


24-835 


113 


35-974 


9 


2.924 


44 


14.018 


79 


25-I53 


114 


36.292 


10 


3- 2 36 


45 


14-335 


80 


25-471 


115 


36.610 


11 


3-549 


46 


I4-653 


81 


25.790 


Il6 


36.929 


12 


3.864 


47 


14.972 


82 


26.108 


117 


37-247 


13 


4.179 


48 


15.291 


83 


26.426 


Il8 


37-565 


14 


4-494 


49 


15.608 


84 


26.744 


119 


37.883 


15 


4.810 


50 


I5-927 


85 


27.063 


I20 


38.202 


16 


5.126 


5i 


16.244 


86 


27.381 


121 


38.520 


17 


5-442 


52 


16.562 


87 


27.699 


122 


38.838 


18 


5-759 


53 


16.880 


88 


28.OI7 


I23 


39-I56 


19 


6.076 


54 


17.200 


89 


2&S35 


124 


39-475 


20 


6.392 


55 


17.516 


90 


28.654 


125 


39-793 


21 


6.710 


56 


17.835 


91 


28.972 


126 


40.111 


22 


7.027 


57 


18.152 


92 


29.290 


127 


40.429 


23 


7-344 


58 


18.471 


93 


29.608 


128 


40.748 


24 


7.661 


59 


18.789 


94 


29.927 


I29 


41.066 


25 


7-979 


60 


19.107 


95 


30-245 


I30 


41.384 


26 


8.297 


61 


I9-425 


96 


30.563 


131 


41-703 


27 


8.614 1 


62 


19-744 


97 


30.881 


132 


42.021 


28 


8.931 


63 


20.062 


98 


3I.200 


1-33 


42-339 


29 


9.249 


64 


20.380 


99 


3I.5I8 


134 


42.657 


30 


9-567 


65 


20.698 


100 


3I-836 


135 


42.976 


3i 


9.884 


66 


21.016 


101 


32.154 


136 


43-294 


32 


10.202 


67 


21-335 


102 


32-473 


137 


43.612 


33 


10.520 


68 


21.653 


103 


32.791 


138 


43-931 


34 


10.838 


69 


21.971 


104 


33-109 


139 


44.249 


35 


11. 156 


70 


22.289 


105 


33-428 


I40 


44-567 


36 


11.474 


7i 


22.607 


106 


33-740 


141 


44.890 


37 


11. 791 


72 


22.926 


107 


34.058 


142 


45.204 


38 


12. no 


73 


23-244 


108 


34-376 


143 


45-522 



Table 5 — Pitch Diameters for Oxe-Ixch Chordal Pitch 

Teeth jrom 4 to 143 

FOR AXY OTHER PITCH MULTIPLY BY THAT PITCH 



TOOTH PARTS 



41 



CHORDAL THICKNESS OF TEETH 
In order to correctly measure the teeth, the chordal thickness must be used, 
as illustrated by Fig. 40. Also as the location of the pitch line on the sides of 




FIG. 40. CHORDAL TOOTH THICKNESS. 



the teeth falls below the pitch line at the center of tooth. The measurement for 
the addendum must also be corrected, if any degree of accuracy is expected. 
Table 6, gives these corrected dimensions for various standard pitches. 



Number 
of 


1 D 


P. 


lV2 


D. P. 


2 D 


. P. 


2 l A 


D. P. 


Number 
of 


Teeth. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


I.5607 


I.0769 


I.0405 


0.7179 


O.7804 


0.5385 


O.6243 


O.4308 


8 


9 


I.5628 


I.0684 


I.0419 


O.7123 


O.7814 


0.5342 


O.6251 


0.4273 


9 


IO 


I-5643 


I.0616 


I.0429 


0.7077 


O.7821 


O.5308 


O.6257 


O.4246 


10 


II 


I-5654 


1-0559 


I.0436 


0.7039 


O.7827 


O.5279 


O.6261 


O.4224 


11 


12 


I-5663 


1-0514 


I.0442 


O.7009 


O.7831 


0.5257 


O.6265 


O.4206 


12 


14 


I-5675 


I.0440 


I.0450 


0.6960 


O.7837 


O.5220 


O.6270 


O.4176 


14 


17 


I.5686 


I.0362 


I -0457 


O.6908 


O.7843 


0.5181 


O.6274 


O.4145 


17 


21 


I.5694 


I.0294 


I.0463 


O.6863 


O.7847 


0.5I47 


O.6277 


O.4118 


21 


26 


I.5698 


I.0237 


I.0465 


O.6825 


O.7849 


O.5118 


O.6279 


0.4095 


26 


35 


I.5702 


I.OI76 


I.0468 


O.6784 


O.7851 


O.5088 


O.6281 


O.4070 


35 


55 


I.5706 


I.OII2 


I.0471 


O.6741 


0.7853 


O.5056 


O.6282 


O.4045 


55 


135 


I-5707 


I.OO47 


I.0471 


O.6698 


0.7853 


O.5023 


O.6283 


O.4019 


135 


Number 
of 


3 D 


. P. 


z l A 


D. P. 


4 D 


. P. 


5 E 


K P. 


Number 
of 


Teeth. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


O.5202 


0.3589 


0.4459 


0.3077 


O.3902 


0.2692 


0.3121 


0.2154 


8 


9 


0.5209 


O.3561 


0.4465 


O.3052 


O.3907 


O.2671 


O.3126 


O.2137 


9 


10 


0.5214 


0.3538 


0.4469 


0.3033 


O.3911 


O.2654 


O.3129 


O.2123 


10 


11 


O.5218 


0.35I9 


0-4473 


O.3017 


0.39I3 


O.2640 


O.3131 


O.2112 


11 


12 


O.5221 


0.3505 


0.4475 


0.3004 


O.3916 


O.2628 


0.3I33 


O.2103 


12 


14 


0.5225 


O.3480 


0.4479 


O.2983 


O.3919 


O.2610 


0.3I35 


O.2088 


14 


17 


O.5228 


0-3454 


0.4482 


0.2961 


0.3921 


O.2590 


0.3I37 


O.2072 


17 


21 


0.5231 


0.343I 


O.4485 


0.2941 


O.3923 


O.2573 


0.3I39 


0.2059 


21 


26 


0.5233 


O.3412 


0.4485 


0.2925 


0.3925 


0.2559 


0.3140 


O.2047 


26 


35 


0.5234 


O.3392 


O.4486 


0.2907 


0.3926 


O.2544 


0.3140 


O.2035 


35 


55 


0.5235 


0.337I 


O.4487 


O.2889 


O.3927 


0.2528 


0.3141 


0.2022 


55 


135 


O.5236 


0.3349 


0.4488 


0.2871 


0.3927 


0.2512 


0.3141 


O.2009 


135 



Table 6 — Chordal Thicknesses and Addenda of Gear Teeth of Diametral Pitch 

Boston Gear Works 



42 



AMERICAN MACHINIST GEAR BOOK 



Number 
of 


6 D. P. 


7 D. P. 


8 D. P. 


9 D. P. 


Number 
of 


Teeth. 


Thickness. 


Addendum 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


O.2601 


O.I795 


O.2230 


O.I538 


0.1951 


O.1346 


O.I734 


O.1197 


8 


9 


O.2605 


O.1781 


O.2233 


O.1526 


O.I954 


O.1336 


O.1736 


0.1187 


9 


IO 


O.2607 


O.1769 


O.2235 


O.1517 


O.I955 


O.1327 


O.1738 


O.1180 


10 


ii 


O.2609 


0.1760 


O.2236 


O.1508 


O.I957 


O.1320 


O.1739 


O.1173 


11 


12 


O.2610 


O.1752 


O.2238 


O.1502 


0.1958 


O.1314 


O.1740 


O.1168 


12 


14 


O.2612 


O.1740 


O.2239 


O.1491 


O.I959 


O.1305 


O.1742 


0.1160 


14 


17 


O.2614 


O.1727 


O.2241 


O.1480 


O.1961 


O.1295 


O.I743 


O.1151 


17 


21 


O.2616 


O.1716 


O.2242 


O.1471 


O.1962 


O.1287 


O.I744 


O.1144 


21 


26 


O.2616 


O.1706 


O.2243 


O.1462 


O.1962 


0.1280 


O.I744 


O.1137 


26 


35 


O.2617 


O.1696 


O.2243 


O.I454 


O.1963 


O.1272 


O.I745 


0.1131 


35 


55 


O.2618 


O.1685 


0.2244 


O.I445 


0.1963 


0.1264 


O.I745 


O.1124 


55 


135 


O.2618 


O.1675 


O.2244 


O.I435 


O.1963 


O.1256 


O.I745 


O.II16 


135 


Number 
of 


10 D. P. 


11 D. P. 


12 D. P. 


13 D. P. 


Number 
of 


Teeth. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


0.1561 


O.1077 


O.1419 


O.0979 


O.1301 


O.0897 


O.I20I 


O.0828 


8 


9 


O.1563 


O.1068 


0.1421 


O.0971 


0.1302 


O.0890 


0.1202 


O.0822 


9 


10 


O.1564 


0.1061 


O.1422 


0.0965 


O.1304 


O.0885 


O.I203 


O.0816 


IO 


11 


O.1565 


O.1056 


O.1423 


O.O960 


O.1305 


O.0880 


O.I204 


O.0812 


II 


12 


O.1566 


O.IO51 


O.1424 


O.0956 


O.1305 


O.0876 


O.I205 


O.0809 


12 


14 


0.1567 


O.IO44 


O.1425 


O.0949 


O.1306 


O.0870 


O.I206 


O.0803 


14 


17 


O.1569 


O.1036 


0.1426 


O.0942 


0.1307 


0.0863 


O.I207 


O.0797 


17 


21 


O.1569 


O.1029 


O.1427 


O.0936 


O.1308 


O.0858 


O.I207 


O.0792 


21 


26 


O.1570 


O.1024 


O.1427 


O.0931 


O.1308 


O.0853 


O.I207 


O.0787 


26 


35 


O.1570 


O.IO18 


0.1427 


O.0925 


O.1309 


O.0848 


O.I208 


O.0782 


35 


55 


O.1571 


O.IOII 


O.1428 


O.0919 


O.1309 


O.0843 


O.I2o8 


O.0777 


55 


135 


O.1571 


0.1005 


O.1428 


O.0913 


O.1309 


0.083 7 


O.I208 


O.0772 


135 


Number 
of 


14 D. P. 


15 D. P. 


16 D. P. 


17 D. P. 


Number 
of 


Teeth. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


O.1115 


O.0769 


O.I040 


O.0718 


O.0975 


O.0673 


O.0918 


O.0633 


8 


9 


0.1116 


O.0763 


0.1042 


O.0712 


0.0977 


O.0669 


O.0919 


O.0628 


9 


10 


O.1117 


O.0758 


O.1043 


O.0709 


O.0978 


O.0664 


O.0920 


O.0624 


IO 


11 


0.1118 


O.0754 


O.IO44 


O.0704 


O.0978 


O.0659 


O.0921 


O.0621 


11 


12 


0.1119 


O.0751 


O.IO44 


O.0701 


0.0979 


O.0657 


O.0921 


O.0618 


12 


14 


O.1119 


O.0746 


O.1045 


O.0696 


O.0980 


O.0652 


O.0922 


O.0614 


14 


17 


0.1120 


O.0740 


O.IO46 


O.0691 


O.0980 


O.0648 


O.0923 


0.0609 


17 


21 


O.1121 


O.0735 


O.1046 


0'o686 


O.0981 


O.0643 


0.0923 


O.0605 


21 


26 


O.II2I 


O.0731 


O.IO46 


O.0682 


O.0981 


O.0640 


O.0923 


O.0602 


26 


35 


O.II22 


O.0727 


O.IO47 


O.0678 


O.0981 


O.0636 


O.0924 


O.0598 


35 


55 


O.II22 


O.0722 


O.1047 


O.0674 


O.0981 


O.0632 


0.0924 


O.0595 


55 


135 


O.II22 


0.0718 


O.1047 


O.0670 


0.0981 


O.0628 


0.0924 


O.0591 


135 



Chordal Thicknesses and Addenda oe Gear Teeth of Diametral Fitch— Continued 

Boston Gear Works 



TOOTH PARTS 



43 



Number 
of 


18 D. P. 


19 D. P. 


20 D. P.. 


24 D. P. 


Number 
of 


Teeth. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


O.0867 


6.0598 


0.0821 


O.0567 


O.0780 


O.0538 


0.0650 


0.0448 


8 


9 


O.0868 


0-0593 


O.0822 


0.0562 


O.0781 


O.0534 


0.0651 


O.0445 


9 


IO 


O.0869 


O.0589 


O.0823 


O.0558 


O.0782 


O.0530 


O.0651 


O.0443 


IO 


II 


O.0869 


O.0586 


0.0824 


0.0555 


0.0783 


O.0528 


0.0652 


0.9439 


II 


12 


O.0870 


O.0584 


0.0824 


0.0553 


O.0784 


O.0525 


O.0653 


0.0437 


12 


14 


O.0871 


O.0580 


0.0825 


O.0549 


0.0784 


O.0522 


O.0653 


0.0435 


14 


17 


O.0871 


O.0575 


O.0826 


0.0545 


O.0784 


O.0518 


O.0653 


O.0432 


17 


21 


O.0872 


O.0572 


0.0826 


0.0542 


0.0785 


O.0514 


0.0654 


0.0429 


21 


26 


O.0872 


O.0568 


0.0826 


O.0538 


O.0785 


O.0511 


O.0654 


O.0426 


26 


35 


O.0872 


O.0565 


0.0826 


0.0535 


0.0785 


O.0508 


0.0654 


0.0424 


35 


55 


O.0873 


O.0562 


0.0827 


O.0532 


O.0785 


O.0505 


O.0654 


O.0421 


55 


135 


O.0873 


O.0558 


0.0827 


0.0528 


0.0785 


O.0502 


O.0654 


0.0419 


135 



Chordal Thicknesses and Addenda of Gear Teeth of Diametral Pitch — Continued 



Number 
of 


w 


C. P. 


M" 


C. P. 


%" 


C. P. 


i" 


C. P. 


Number 
of 


Teeth. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


O.3105 


O.2142 


0.3725 


O.2570 


0.4347 


O.2997 


0.4968 


O.3426 


8 


9 


O.3109 


0.2125 


O.3730 


O.2550 


0.4353 


O.2976 


0.4974 


O.3400 


9 


10 


O.3112 


0.2II2 


0.3734 


•02534 


0.4357 


O.2957 


O.4978 


0.3378 


IO 


11 


O.3114 


0.2IOO 


0.3737 


O.2520 


O.4360 


0.2941 


0.4982 


0.3360 


II 


12 


O.3116 


O.209I 


0-3739 


O.2510 


O.4363 


O.2938 


O.4986 


0.3346 


12 


14 


O.3118 


O.2077 


0.374I 


O.2492 


O.4366 


O.2908 


O.4988 


O.3322 


14 


17 


O.3120 


0.206l 


0-3744 


O.2473 


O.4369 


O.2886 


O.4992 


0.3298 


17 


21 


O.3122 


O.2048 


O.3746 


O.2457 


0.437I 


O.2868 


O.4994 


0.3276 


21 


26 


O.3123 


O.2036 


O.3748 


O.2443 


O.4372 


O.2851 


0.4997 


O.3258 


26 


35 


O.3124 


0.2024 


O.3748 


O.2429 


0.4373 


O.2833 


O.4999 


0.3238 


35 


55 


O.3124 


0.20II 


0.3748 


O.2414 


0.4374 


O.2816 


0.4999 


O.3218 


55 


135 


O.3124 


O.I999 


O.3748 


O.2398 


0.4374 


O.2798 


O.4999 


O.3198 


135 


Number 
of 


Ifc" 


C. P. 


iH" 


C. P. 


l%" 


C. P. 


2" 1 


2. P. 


Number 
of 


Teeth. 


Thickness. 


Addendum. 


Thickness. 


Addendum, 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


O.6210 


O.4284 


O.7450 


O.5140 


O.8694 


0.5994 


O.9936 


O.6852 


8 


9 


O.6218 


O.4250 


O.7460 


O.5100 


O.8706 


0.5952 


O.9948 


O.6800 


9 


IO 


O.6224 


O.4224 


O.7468 


0.5068 


O.8714 


O.5914 


0.9956 


0.6756 


IO 


II 


O.6228 


O.4200 


0.7474 


O.5040 


O.8720 


0.5882 


O.9964 


O.6720 


II 


12 


O.6232 


O.4182 


O.7478 


O.5020 


O.8726 


O.5876 


O.9972 


O.6692 


12 


14 


O.6236 


O.4154 


O.7482 


O.4984 


O.8732 


O.5816 


O.9976 


O.6644 


14 


17 


O.6240 


O.4122 


O.7488 


O.4946 


O.8738 


0.5772 


O.9984 


O.6596 


17 


21 


O.6244 


O.4096 


O.7492 


0.4914 


O.8742 


0.5736 


O.9988 


0.6552 


21 


26 


O.6246 


O.4072 


O.7496 


0.4886 


O.8744 


O.5702 


0.9994 


0.6516 


26 


35 


O.6248 


O.4048 


O.7498 


O.4858 


O.8746 


0.5666 


O.9998 


O.6476 


35 


55 


O.6250 


O.4022 


O.7499 


O.4828 


O.8748 


O.5632 


O.9999 


O.6436 


55 


135 


O.6250 


O.3998 


O.7499 


0.4796 


0.8748 


0.5596 


0.9999 


O.6396 


i35 



Table 7 — Chordal Thicknesses and Addenda of Gear Teeth of Circular Pitch 

Boston Gear Works 



44 AMERICAN MACHINIST GEAR BOOK 

INVOLUTE CUTTERS 

Until quite recently involute cutters were made in sets of eight, as follows: 

Number 
of Cutter 

i for 135 teeth to rack 

2 for 55 to 134 teeth 

3 for 35 to 54 teeth 

4 for 26 to 34 teeth 

5 for 21 to 26 teeth 

6 for 17 to 20 teeth 

7 for 14 to 16 teeth 

8 for 12 to 13 teeth 

Modern conditions, however, require a more accurate tooth than can be 
produced by this number of cutters. A set of fifteen, utilizing the half numbers 
is now in common use. 



Number 
of Cutter 

i for 135 teeth to a rack 

\Y2 for 80 to 134 teeth 

2 for 55 to 79 teeth 
2 X A for 42 to 54 teeth 

3 for 35 to 41 teeth 
3^ for 30 to 34 teeth 

4 for 26 to 29 teeth 
4H for 23 to 25 teeth 

5 for 21 to 22 teeth 
5^ for 19 to 20 teeth 

6 for 17 to 18 teeth 
63^ for 15 to 16 teeth 



7 for 14 teeth 
7K for 13 teeth 

8 for 12 teeth 

To produce accurate gears, templets for tooth thickness, made according 
to Tables 6 and 7, should be used instead of using one templet for each pitch 
and depending upon the workman's judgment as to how much shake to allow 
for different numbers of teeth. These templets, made up according to Tables 
6 and 7, which are based on the use of eight cutters for each pitch, should be 
sufficiently accurate for all practical purposes. 



SECTION II 

Spur Gear Calculations 

To find the pitch diameters of two gears, the number of teeth in each and 
"the distance between centers being given: Divide twice the distance between 
centers by the sum of the number of teeth: Find the pitch diameter of each 
gear separately by multiplying this quotient by its number of teeth. 

Example: Find the pitch diameters of a pair of spur gears, 21 and 60 teeth, 

for 25-inch centers. 

2 X 25 , 
7-jr- = O.617284, 

21 + 60 

0.617284 X 21 = 12.96296 inches, or the pitch diameter of the pinion 
0.617284 X 60 = 37.03704 inches, or the pitch diameter of the gear 

The distance between the centers is one-half the sum of the pitch diameters. 
In the above example the center distance would prove to be: 

_IM6£96±17i>37°£ =25inches 



NOTATIONS FOR FORMULAS 

p = diametral pitch 



D r = pitch diameter 
D = outside diameter 
V = velocity 

d f = pitch diameter 
d = outside diameter 
v = velocity 



gear 



pinion 



These gears run together 



a = distance between the centers 
b = number of teeth in both 



45 



4 6 



AMERICAN MACHINIST GEAR BOOK 



TO FIND 



HAVING 



N 



N 



n 



D' 



a and p 
D' and &' 

b and p 
n v and V 

b v and V 

b v and V 

N v and V 

pD' V and v 

N V and n 

pD' V and n 

n v and N 
a v and V 
a v and V 



RULE 



The continued product of center dis- 
tance, pitch and 2 



One-half the sum of the pitch di- 
ameters 



Divide the total number of teeth by 
twice the pitch 



Divide the product of the number of 
teeth and velocity of pinion by the 
velocity of gear 



Divide the product of the total num- 
ber of teeth and velocity of pinion 
by the sum of the velocities 



Divide the product of the total num. 
ber of teeth and the velocity of gear 
by the sum of the velocities 



Divide the product of the number of 
teeth in gear and its velocity by 
the velocity of pinion 



Divide the continued product of the 
pitch, pitch diameter and velocity 
of the gear by the velocity of pinion. 

Divide the product of the number of 
teeth and velocity of gear by the 
number of teeth in pinion 



Divide the continued product of the 
pitch, pitch diameter and velocity 
of gear by the number of teeth in 
pinion 



Divide the product of the number of 
teeth in pinion and its velocity by 
the number of teeth in gear 



Divide the continued product of the 
center distance, velocity of pinion 
and 2, by the sum of the velocities. 

Divide the continued product of the 
center distance, velocity of gear and 
2, by the sum of the velocities 



FORMULA 



a p 2 

D' + d' 

2 

b 

2p 

n v 
~V~ 

b v 
v+ V 



NV 



pD'V 



NV 



pD'V 



n v 



N 



2 a v 




v+V 



EXAMPLE 



15 X 3 X 2 =90 

20 + IO 

15 



= 60 



2 


90 
2X3 


30 X 2 


I 


90 X 2 


2 + 1 


90 X 1 


2 + 1 " 


60 X 1 



= 60 



30 



= 30 



3 X 20 X 1 



30 



60 X 1 
30 



= 2 



3 X 20 X 1 





30 






3^X2 




60 




2 


X 15 x 

2 + 1 


2 


2 


X 15 X 


I 



= 2 



2 + 1 



= I 



= 20 



= IO 



Table 8 — Formulas for a Pair of Mating Spur Gears 



SPUR GEAR CALCULATIONS 



47 



TO FIND 



The Diametral 
Pitch. 

The Diametral 
Pitch. 

The Diametral 
Pitch. 



HAVING 



Pitch 



Diameter 



Pitch 



Diameter 



Pitch 



Diameter. 

Pitch 

Diameter. 

Outside 

Diameter. 

Outside 

Diameter. 

Outside 

Diameter. 

Outside 

Diameter. 

Number of 

Teeth. 

Number of 

Teeth. 

Thickness 

of Tooth. 

Addendum. 



Dedendum. 
Working 

Depth. 

Whole Depth, 

Clearance. 
Clearance. 



The Circular Pitch.. 

The Pitch Diameter 
and the Number of 
Teeth 

The Outside Diame 
ter and the Number 
of Teeth 

The Number of Teeth 
and the Diametral 
Pitch 

The Number of Teeth 
and Outside Diam- 
eter 

The Outside Diame- 
ter and the Diame 
tral Pitch 

Addendum and the 
Number of Teeth 

The Number of Teeth 
and the Diametral 
Pitch 

The Pitch Diameter 
and the Diametral 
Pitch 

The Pitch Diameter 
and the Number of 
Teeth 

The Number of Teeth 
and Addendum . . 

The Pitch Diameter 
and the Diametral 
Pitch 

The Outside Diame 
ter and the Diame 
tral Pitch 



RULE 



The Diametral Pitch, 
The Diametral Pitch. 

The Diametral Pitch. 
The Diametral Pitch. 

The Diametral Pitch. 

The Diametral Pitch. 
Thickness of Tooth.. 



Divide 3.1416 by the Circular 
Pitch 



Divide Number of Teeth by- 
Pitch Diameter 



Divide Number of Teeth plus 2 
by Outside Diameter 



Divide Number of Teeth by the 
Diametral Pitch 



Divide the product of Outside 
Diameter and Number of Teeth 
by Number of Teeth plus 2 . . . 

Subtract from the Outside Diam- 
ter the quotient of 2 divided by 
the Diametral Pitch 

Multiply Addendum by the 
Number of Teeth 



Divide Number of Teeth plus 2 
by the Diametral Pitch 



Add to the Pitch Diameter the 
quotient of 2 divided by the 
Diametral Pitch 

Divide the product of the Pitch 
Diameter and Number of Teeth 
plus 2 by the Number of Teeth 

Multiply the Number of Teeth 
plus 2 by Addendum 



Multiply Pitch Diameter by the 
Diametral Pitch 



Multiply Outside Diameter by 
the Diametral Pitch and sub- 
tract 2 

Divide 1.5708 by the Diametral 
Pitch 

Divide 1 by the Diametral Pitch, 

PL 

N 

Divide 1.157 by the Diametral 
Pitch 



or s 



Divide 2 by the Diametral Pitch. 

Divide 2.157 by the Diametral 
Pitch 

Divide 0.157 by the Diametral 
Pitch 

Divide Thickness of Tooth at 
pitch line by 10 





FORMULA 


p 


3.1416 
P' 


p 


N. 
~ D' 


p 


N + 2 
D 


D' 


N 
P 


U 


D N 



N + 2 
D' = D-- 

D' = sN 

N + 2 



D = 



D = D> + — 



D = 



(iV+ 2) D' 



N 

D = (N+2) s 
N = D' p 

N = Dp - 2 



1 
P 
s+f- 



W = 



i-i57. 



W'+f = 



2-157 



0.157 



f -~™ 



Table 9— Spur Gear Calculations for Diametral Pitch 

14^ Degree Standard 
R. D. Nuttall Company 



4 8 



AMERICAN MACHINIST GEAR BOOK 



TO FIND 



HAVING 



The Circular 
Pitch. 

The Circular 
Pitch. 

The Circular 
Pitch. 



Pitch 



Diameter. 



Pitch 



Diameter. 



Pitch 



Diameter. 

Pitch 

Diameter. 

Outside 

Diameter. 

Outside 

Diameter. 

Outside 

Diameter. 

Number of 

Teeth. 

Thickness 

of Tooth. 

Addendum. 



Dedendum. 

Working 

Depth. 

Whole Depth. 

Clearance. 

Clearance. 



The Diametral Pitch. 

The Pitch Diameter 
and the Number of 
Teeth 

The Outside Diame- 
ter and the Number 
of Teeth 

The Number of Teeth 
and the Circular 
Pitch 

The Number of Teeth 
and the Outside Di- 
ameter 

The Outside Diame- 
ter and the Circular 
Pitch 

Addendum and the 
Number of Teeth. . 

The Number of Teeth 
and the Circular 
Pitch 

The Pitch Diameter 
and the Circular 
Pitch 

The Number of Teeth 
and the Addendum 

The Pitch Diameter 
and the Circular 
Pitch 

The Circular Pitch. . . 
The Circular Pitch. . . 



The Circular Pitch. . 
The Circular Pitch. . 
The Circular Pitch. . 
The Circular Pitch. . 
Thickness of Tooth. 



RULE 



FORMULA 



Divide 3.1416 by the Diametral 
Pitch I 

Divide Pitch Diameter by the! 
product of 0.3183 and Number 
of Teeth 

Divide Outside Diameter by the 
product of 0.3183 and Number 
of Teeth plus 2 I 

The continued product of the 
Number of Teeth, the Circular 
Pitch and 0.3183 

Divide the product of Number of 
Teeth and Outside Diameter 
by Number of Teeth plus 2 . . . j 

Subtract from the Outside Diam- 
eter the product of the Circular 
Pitch and 0.6366 

Multiply the Number of Teeth by 
the Addendum 



P' = 
P'=, 



3-i4i6 

P 

D' 

0.3183 N 

D 



0.3183 N+2 
D' = Xp' 0.3183 



D' = 



XD 

N+2 



The continued product of the 
Number of Teeth plus 2 the 
Circular Pitch and 0.3183 .... 

Add to the Pitch Diameter the 
product of the Circular Pitch 
and 0.6366 

Multiply Addendum by Number 
of Teeth plus 2 

Divide the product of Pitch Di- 
ameter and 3.1416 by the Cir 
cular Pitch 



One-half the Circular Pitch . 



Multiplv the Circular Pitch by 
0.3183, OT s = 



N 

Multiplv the Circular Pitch by 
0.3683 

Multiply the Circular Pitch by 
0.6366 

Multiplv the Circular Pitch by 
0.6866 

Multiply the Circular Pitch by 



0.0: 



One-tenth the Thickness of Tooth 
at Pitch Line 



D' = D- 

(P' 0.6366) 

D' = Ns 

D = (X+2) 

p' 0.3183 

D = D + 

(p' 0.6366) 

D = s (X + 2) 

1X3.1416 
P 



t = 



P' 



s = p' 0.3183 

* + /- = // 0.3683 

W = p' 0.6366 
W = p' 0.6S66 
f = p 0.05 



f = 



10 



Table 10 — Spur Gear Calculations for Circular Pitch 

14^ Degree Standard 

R. D. Xuttall Company 



SECTION III 

Speeds and Powers 

transmission of power by gearing with particular reference 

to spur and bevel gears 

SPEED RATIO 

The problem of finding the proper diameter or speed of a gear or pulley is 
simple enough when once thoroughly understood. 

The gear may be represented by its number of teeth, pitch diameter, pitch 
radius, or speed ratio, as the case may be. In the explanation to follow the 
number of teeth is used. The speed is in revolutions per minute. 

Rule: Divide the product of the speed and number of teeth of one gear by 
the speed or number of teeth of its mate to secure the lacking dimension. 

That is, if both the speed and number of teeth are known for one gear, mul- 
tiply the speed by the number of teeth, and divide this product by the known 
quantity of the mating gear to secure its number of teeth or speed, as the case 
may be. 

Or the same result may be obtained by proportion, the values being placed 
as follows: 

n : N : : R : r (i) 

n = number of teeth in pinion 
r = revolutions per minute of pinion 
N = number of teeth in gear 
R = revolutions per minute of gear 

Example: A gear having 60 teeth makes 300 revolutions per minute, what 
will be the speed of an engaging pinion having 1 5 teeth? 

n : N : R : r 
15 : 60 : 300 : x 

Therefore, x = , or 1 200 revolutions per minute for pinion n 

To compute these values for a train of gears, use the continued product of 
the pinions and the continued product of the gears as a single gear and pinion 
and proceed as above. 

49 



5° 



AMERICAN MACHINIST GEAR BOOK 



Example: In Fig. 41, the gear N has 100 teeth, N', 70 teeth, N" , 60 
teeth, n, 15 teeth; n', 18 teeth; and n" ', 24 teeth. The gear N makes 10 revolu- 
tions per minute. What will be the speed of the pinion n"? 

N, N' and N" = 100 X 70 X 60 = 420,000 
n, n f and / = 15 X 18 X 24 = 6,480. 
n : N : : R : r 
6.480 : 420,000 : : 10 : x 

420,000 X 10 



Therefore, x = 



6,480 



, or 648 revolutions per minute for pinion n" . 




FIG. 41. GEAR TRAIN. 



FIG. 42. INTERMEDIATE GEAR DOES 
NOT AFFECT THE SPEED RATIO. 



The velocities of a train of gears may also be found as follows: N, N f , N fJ 
and n, n', n" ', etc., representing the number of teeth in the gears and pinions. 

RN N' N" 



r = 



R = 



n n' n" 
r nn f n n 

N N' N" 



(2) 
(3) 



The intermediate gear B, as shown in Fig. 42, while it changes the direction 
of the rotation of the gears, A and C does not alter their speed ratio, the cir- 
cumferential velocities of all three gears being equal. 



ARRANGEMENT OF GEAR TRAINS 

For compound reduction there must be four gears, as per Fig. 43, the gears 
B and C being keyed to an intermediate shaft, the power being transferred to 
the machine by the shaft-carrying gear D. 

When a great reduction is required, say 64 to 1, there may be two inter- 
mediate shafts, as in Fig. 44. 

This reduction might be accomplished by using a drive, as in Fig. 43, divid- 
ing the total reduction between two sets of gears, but a triple reduction is used 
by way of illustration. The best results are always obtained by dividing the 
reduction as evenly as possible among the different pairs of gears. For instance: 
for a double reduction, as in Fig. 43, the ratio of each pair should be made as 



SPEEDS AND POWERS 



51 



near the square root of the total reduction as possible. In case of the triple 
reduction, Fig. 44, the ratio of each pair should be the cube root of the total 

reduction, or v 64 = 4. That is, there are three sets of gears, each having a 
speed ratio of 4 to 1. If double reduction had been used the reduction of each 

gear would have been V 64 = 8, or two sets of gears each having a speed ratio 
of 8 to 1. 

Gear trains proportioned in this way give the highest possible efficiency. 
For instance: an unsuccessful single gear reduction of 16 to 1 might be 



4 



B 



Motor 



flfl 



hww 



4 



Motor 






D 



FIG. 



43. DOUBLE GEAR 
REDUCTION. 



PIG. 



44. TRIPLE GEAR 
REDUCTION. 



made efficient by substituting two pairs of gears, each having a ratio of 4 to 1. 
Making the compound gears 8 to 1 and 2 to 1 would help, but would not be as 
efficient as the equal reduction. This will be especially noticeable in long leads 
in the lathe or milling machines. 



POWER RATIO 

The relative powers of a train of gears are inversely proportional to their 
circumferential velocities. The circumferential velocity of each pair of gears 
in a train being equal, the driving pinion, as shown in Figs. 45 and 46, is ig- 
nored in the calculations for a single pair, the circumferential velocity and the 
load on the teeth being the same as for the mating gear. The problem is 
to determine the power ratio between the drum r and the gear R. 



52 



AMERICAN MACHINIST GEAR BOOK 



Ignoring friction, the values of this drive may be found by proportion, ar- 
ranged as follows: 

W : R : : F : r (4) 

Enough must be added to the load W or taken from the effective lifting force 
F to overcome the frictional resistance of the teeth and bearings. This loss 
must be estimated and the percentage of loss added to the load W, the ratio of 
R and r being determined according to this new ratio. 




FIG. 45. FIG. 46. 

POWER RATIO DIAGRAMS. 



Example: Referring to Fig. 45: if the radius of the gear R is 18 inches, the 
radius of the drum r three inches, what power will be required at F to raise 
300 pounds at W? 

W : R : : F : r 

300 : 18 : : x : 3 

300 X 3 



Therefore, x = 



18 



50 pounds required at F. 



Suppose the loss in efficiency to be 10 per cent and the radius of the gear R 
18 inches. What must be the radius of the drum r to raise 300 pounds at W? 

300 + 10 per cent = 330 pounds. 

W : R : : F : T 

300 : 18 : : 50 : x 

18 X 50 



Therefore, x = 



33o 



= 2.7 inches for the radius of drum r. 



For a train of gears, the continued products of the driving and driven gears 
may be considered as single gears. Or the power ratio may be considered be- 
tween each pair inversely proportional to their velocity ratios. 



SPEEDS AND POWERS 



53 



Example: Referring to Fig. 47; what force is required at F to raise 2500 
pounds at W, the loss in efficiency being 30 per cent? 

R R' R" = 20 X 18 X 10 = 3600 
rr'r" = 6 X 8 X 5 = 240 

W = 2500 + 30 per cent = 3250. 



W : R V F •' r 3*50X240 A „ 

3250 : 3600 : : x : 240, x = — , or 217 pounds at F. 




A1 _ W rr f r" 3250 X 6 X 8 X 5 , 

Also F = ie r> R>>> = 20 x 18 x 10 "' = 2I7 P° unds - 


(5) 


. , TT7 FRR'R' f 217X20X18X10 


(6) 


rr'r"' 6X8X5 



AN EXAMPLE IN HOIST GEARING 

Example: What gears will be required to lift a load of 2400 pounds at a 
uniform rate of speed, employing a 10 horse-power motor running n 20 revolu- 
tions per minute, driving with a rawhide pinion 4 inches pitch diameter? See 
Fig. 48. 

F = 281 Pounds 



1120 
R.P.M 





1173 Feet 
per Min. 



FIG. 47. POWER RATIO OF GEAR TRAINS. 



2400Pounds- 



FIG. 48. EXAMPLE OF GEAR DRIVE FOR HOIST. 



Velocity of pinion in feet per minute, V = d' 0.2618 R. P. M. (7) 

HP X 33,000 



The safe load, W + 



(8) 



Therefore, V = 4X 0.2618 X 1120= n 73 feet per minute. 

T O ^^ 3 'Z OOO 

And W = — , or 281 pounds, which is the load to be carried by 

"73 
the pinion. 



54 AMERICAN MACHINIST GEAR BOOK 

Assuming that 20 per cent is lost by the friction of the gear teeth, bearings, 
etc., the real load to be raised by the force of 280 pounds at the pitch line of 
the driving pinion is: 

2400 + 20 per cent. = 2880 pounds. 

The necessary velocity ratio of the gears to equal this ratio of power is, 
therefore : 

2880 10.25 

~rtiT> ~ 1 " 

This reduction must be made between R' and r", and R and r', the pinion 
r not being considered as its velocity is the same as that of the gear R, there- 
fore the load on the teeth will be the same. 

Since it is always best to make the reduction in even steps, and double re- 
duction is desirable for a ratio of 10.25 to 1 take the square root of the total 
reduction, 10.25, which is approximately 3.2 to 1 for each reduction. Prac- 
tically, however, a reduction of ^ — - and — ^— will answer. 

The ratio between R r and r" is made — — . Assuming the diameter of the 

drum r" to be 10 inches, the pitch diameter of the gear R! will be 3.4 X 10 = 34 

inches. The ratio between R and r' is -*— , assuming the pitch diameter of the 

pinion r' to be 7 inches, the pitch diameter of the gear R will be 7 X 3 = 21 
inches. 

The power or circumferential force of the gear R is, of course, that of the 
driving pinion r, 281 pounds. Therefore, the power of the pinion r', and con- 
sequently that of the gear R', is 281 X 3 = 843 pounds. 

The problem is now reduced to two simple ones, that is — to design a pair of 
gears r and R to transmit a force of 281 pounds at a speed of n 73 feet per 
minute, and a second pair r' and R' to transmit a force of 843 pounds at a speed 
of 390 feet per minute. 

It is necessary to assume a pitch judged to be suitable for the different drives 
and to try its value for carrying the required load by the Lewis formula, ob- 
taining the safe load per inch of face, and make the face sufficiently wide to 
transmit the power. 

For the first pair of gears, r and R, assume 4 diametral pitch — 0.7854-inch 
circular pitch — allowing 5000 pounds per square inch as a safe stress for raw- 
hide. Number of teeth in pinion r = 4 X 4 = 16. 

Safe load per inch of face = spy — , T . . (See formula 24.) 

600 + V 



SPEEDS AND POWERS 55 

Or 5000 X 0.785 X 0.077 T — "V =100 pounds per inch of face. 

Making the face of the gears r and R 3 inches it will safely carry 3 X 100 = 
300 pounds, which is sufficient. 

For the second pair, r' and R' ', try 3 diametral pitch — 1.0472-inch circular 
pitch — both gears of cast iron. Figure the strength of the pinion, as it is the 
weaker of the two. Allow 8000 pounds per square inch as a safe stress. 

For a pinion 7 inches pitch diameter, 3 pitch, the number of teeth equals 
7X3= 21 teeth. Factor y for 21 teeth equals 0.092. W — 8000 X 1.047 X 

0.092 — — . — : — , or 467 pounds per inch of face. 
600 + 390 

Making the face 3 inches, the gears will carry a load of 3 X 467 = 1401 
pounds. These gears will therefore be heavier than necessary, but owing to 
the nature of the service this should be the case, especially as they are made 
of cast iron. 

From the ratio of this train of gears it will be found that the load will be 

1173 
raised at ■ — —~ — = 115 feet per minute, using the full speed of the motor. If 
3 X 3.4 

the load must be raised at a greater speed than 115 feet, a more powerful motor 
would be required, and if at a lower speed there must be a greater gear re- 
duction. For instance, if the hoisting speed had been 80 feet per minute the 

speed ratio would be — -^- = , instead of — — as in the example. 

r 80 1 1 

The above problem is generally put before the designer in a different manner 
— that is the load and speed at which the load is to be raised are given, the 
size of motor and ratio of gearing, etc., to be determined. 

Example: A load of 2400 pounds is to be raised at the uniform rate of 115 
feet per minute; what size motor and what gears will be required? 

Assuming as before a loss of 20 per cent in efficiency in the driven gears, 
bearing, etc., this load will require: 

-, = 10 horse power (2400 pounds + 20 per cent = 2880 pounds). 

33,000 

Using a rawhide pinion four-inch pitch diameter on the motor, we consider 

the problem in the same manner as in previous examples making the ratio of 

„ 2880 10.25 

the gears; — - — = — 

281 1 

The problem of determining the proper gears is the same. 



56 



AMERICAN MACHINIST GEAR BOOK 



RAILWAY GEARS 



Speed in feet per minute at rim of car wheel V = 88 X speed of car in 
miles per hour. (9) 

Speed in feet per minute at pitch line of gear V = 88 X miles per 
hour X R. (10) 



Ratio of gear to wheel 



pitch diameter of gear 
diameter of wheel 



V 



V 



(11) 



-17 , •. 1 r * 77 HP X 33>°°° Kw X 44,102 . . 
Force at pitch line of gear F = ^ , or — ~ — • ( I2 ) 



Fiber stress in tooth 



S = , j. 600 , (See formula 18.) (13) 

P J y 600 + V 



Traction effort at wheel T = 



Horse power HP = 



Speed of car in miles per hour M = 



Traction effort 



T = 



F or Pressure at Pitch 
Line=3553 Pounds. 




Tractive Effort = 5040 Pounds. 

From Formula 12, F= 5040 x 26-8 =3550 Pounds 
33 

FIG. 49. RAILWAY GEARS. 



R' 

TM 

°-375 ' 

Dia. of wheel X teeth in pinion X 
revolution per minute of pinion. 

teeth in gear X 336 

teeth in gear X 24 X gear efficiency 

X torque of motor 

M X diameter of wheel 



(14) 



ds) 



(16) 



(17) 



Example: A car weighing 60 tons 
driven by four motors accelerating at 
the rate of i}4 miles per hour, per 
second, reaches the peak of its start- 
ing torque when at a speed of 20 
miles per hour. The gears are 20 and 
67 teeth 2}4 diametral pitch (1.26 
inches circular pitch) 5M inch face. 
The diameter of the car wheels is 38 
inches. It is required to know the 
maximum fiber stress in pinion tooth. 
The power exerted by motors at its 
peak is 400 kilowatts (800 amperes at 
500 volts). See Fig. 49. 



SPEEDS AND POWERS 



57 



Kilowatts per motor 
Pitch diameter of gear 
Ratio of gear to wheel 



Kw = = ioo Kw 

4 

67 



D' = 



2y 2 



26.8" 



j? 26 - 8 

R = --^-= 0.705 



38 



Speed of gear in feet per minute 



Force at pitch line 

Fiber stress in pinion tooth 5 = 



V = 88 X 20 X 0.705 = 1 241 feet per minute. 

100 X 44.102 
b = - — = 3553 pounds. 



1241 



3553 



= 18,300 

— 7 pounds 

600 ^ 

1.26 X 5.25 X 0.00 X 7 : per square 

600+1241 inch> 



STRENGTH OF GEAR TEETH 

Lewis 
W = load transmitted in pounds (same as value F), 
p' = circular pitch, 
/ = face, 

y = factor for different numbers and forms of teeth (Table 11), 
S = safe working stress of material, 
V = velocity in feet per minute, 

w - s ^y'-l^Tr (18) 



NUMBER 


VALUE OF FACTOR y 


NUMBER 


VALUE OF FACTOR y 














OF 
TEETH 


INVOLUTE 
20° 


INVOLUTE 

15° 

CYCLOID AL 


RADIAL 
FLANKS 


OF 
TEETH 


INVOLUTE 
20° 


INVOLUTE 

15° 

CYCLOID AL 


RADIAL 
FLANKS 


12 


O.078 


O.067 


O.052 


27 


O.III 


O.IOO 


O.064 


13 


O.083 


O.070 


0-053 


30 


0.114 


O.I02 


O.065 


14 


O.088 


O.072 


O.054 


34 


0.118 


O.IO4 


O.066 


15 


O.O92 


0.075 


O.055 


38 


0.122 


O.IO7 


O.067 


16 


O.O94 


O.077 


O.056 


43 


0.126 


O.IIO 


O.068 


17 


O.O96 


O.080 


O.057 


5o 


0.130 


O.II2 


O.069 


18 


O.O98 


O.083 


O.058 


60 


0.134 


O.II4 


O.070 


IQ 


O.IOO 


O.087 


O.059 


75 


0.138 


O.Il6 


O.071 


20 


0.102 


O.O9O 


O.060 


IOO 


0.142 


O.Il8 


O.072 


21 


0.104 


O.092 


O.061 


150 


0.146 


O.I20 


O.O73 


23 


0.106 


O.O94 


O.062 


300 


0.150 


O.I22 


O.074 


25 


0.108 


O.097 


O.063 


Rack 


0.154 


O.I24 


O.075 



Table ii — Values of Factor y for Lewis Formula 



58 AMERICAN MACHINIST GEAR BOOK 

Safe working stress S for 0.30 carbon steel = 15000 

Safe working stress 5 for 0.50 carbon steel = 25000 

Safe working stress S for cast iron = 8000 

Safe working stress S for rawhide = 5000 

AVERAGE VALUES FOR S 

Mr. Lewis' formula for the strength of gears originally read: W — S p' f y, 
a table being given in which the allowable stress of the material 5 was reduced 
as the speed of the gear was increased as follows: 



SPEED OF TEETH IN 
FEET PER MINUTE 


IOO OR 

LESS 


200 


300 


600 


900 


I200 


180O 


24OO 


Cast Iron 

Steel 


8,000 
20,000 


6,000 
15,000 


4,800 
12,000 


4,000 
10,000 


3,000 
7,500 


2.4OO 
6,000 


2,000 
5,000 


1,700 
4,300 



Safe Working Stresses S in Pounds Per Square Inch for Different 

Speeds 



Later Carl G. Barth introduced an equation, , — =7, which gives prac- 

600 + V 

tically the same result as the table when added to the formula, the value S 

being the safe working stress per square inch of the material used, or 

Mr. Barth's equation is the one commonly accepted. It is evident, however, 
that this value will vary for different conditions, the design and workmanship 
being important factors in its proper determination. 

The load is reduced as the speed increases on account of impact. It is evident 
that an accurately spaced and generated gear should have a much higher value 
than one cut by ordinary methods. 

It is also evident that helical and herringbone gears, owing to the nature of 
their tooth contact, should have a much higher value, as they operate under 
entirely different conditions, therefore are capable of heavier loads at higher 
speeds for the same area of tooth contact. Rawhide gears should also have a 
higher factor, as rawhide will absorb shocks that would affect harder materials. 

In the absence of all vibration, and with an indeflectable material, this equa- 
tion could be eliminated from the formula for strength and wear. These are 
conditions that can never be attained, but it is evident that this value will 
stand extended investigation. 



1 • 

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60 AMERICAN MACHINIST GEAR BOOK 

Table 13 was prepared by Henry Hess and published in American Machin- 
ist to allow the pitch and the face to be found with very little arithmetic. 
It is based on Lewis' method of calculating the strength of gear teeth. 

As the bulk of gears in use are either 1 5 degrees involute or cycloid the table 
has been made up of these forms. 

The arrangement of the table is such that, given the number of teeth in 
the gear, and the quotient obtained from dividing the working load per tooth 
by the greatest fiber stress, the pitch and face width can be directly picked out. 
The face width is given in inches and also as a ratio to the circular pitch; it is 
usual to make the face from two or three times the circular pitch — 2.5 is a fair 
average. For overhung gears 2 to 2.5 is proper; for gears supported on both 
sides 2.5 to 3 is good practice. For convenience sake these ratios are repeated at 
the tops of the vertical columns, while the face widths to the nearest sixteenth 
higher corresponding to each ratio for each pitch are given at the foot of the 
various columns. 

As diametral pitches are generally used for light work the table is arranged 
for these from 8 to 3.5, and for heavier work for circular pitches from 1 to 4 
inches. The equivalent circular and diametral pitches are marked in lighter 
faced type. 

Directions and examples are given on page 61. 

The formula from which the tabular values were determined is 



k= Y = P *r (0.124-—), 



which is but another way of writing the Lewis formula, which, with the notation 
changed to agree with formula above, is 



W=S P 'f(o, 24 -°-^), 



the quantity in the parenthesis being the Lewis variable for cycloidal and 15 
degrees involute teeth. The change in form is made by introducing for / its 
value — p' r — as per notation below: 

W = working load in pounds. 

5 = greatest fiber stress in pounds per square inch. 

p' = circular pitch in inches. 

/ = face width in inches. 

n = number of teeth in pinion. 

f 
r = ratio of face width to circular pitch = -^y. 

p = diametral pitch. 



SPEEDS AND POWERS 6 1 

To find the circular pitch p' or diametral pitch p and the face width f: 

Divide the known load W by the permissible greatest fiber stress S; find the 
resulting value k in the body of the table in line with the number of teeth 
in the pinion. Use the pitch given at the top and the face width given at 
the foot of the column. 

Example: A gear of eighteen teeth is loaded with 495 pounds per tooth; the 

permissible fiber stress is 3,000 pounds per square inch. Then — =- = = 

o 3,000 

0.165. Opposite eighteen teeth find k = 0.167 under a diametral pitch 3.5 
and over a face width of 2.25". Or find k = 0.166 under a circular pitch of 
1 inch and over a face width of 2 inches. Either solution will do. 

To find the greatest fiber stress S: 

In the column headed with the pitch used and marked with the face width 
used at the foot, find opposite the tooth number a constant. Divide this into 
the working load imposed on the tooth to get the greatest fiber stress. 

Example: The working load on a tooth of 4-inch circular pitch and 10-inch 
face in a 100-tooth gear is 28,000 pounds. Opposite 100 teeth, under p' = 4 
inches and over/ = 10 inches, find 4.684. Dividing this into the load gives 

28,000 



4.684 



= 5,970 pounds per square inch greatest fiber stress. 



Table 14 for the working strength of gear teeth has been furnished by the 
New Britain Machine Company. It is based on the Lewis formula, and, unlike 
other tables, gives the strength of the teeth when their size is indicated by 
diametral pitch. But one width of face is given for each pitch, this face 
being, as near as may be, that used by the makers of standard gears for the 
market. 

The figures in the body of the table give the working load in pounds for a 
speed of 100 feet per minute for cast-iron gears of the pitch and face found at 
the top of the corresponding column and of the number of teeth given at the 
left of the corresponding line. The left-hand figure at the top of each column 
gives the diametral pitch and the right-hand figure the face in inches. 

600 
For higher speeds the loads are to be reduced by the equation - — —-y 

used as a multiplier, and for steel gears the loads may be increased in proportion 
to the safe load for that material. 



62 



AMERICAN MACHINIST GEAR BOOK 



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SPEEDS AND POWERS 



65 



FACTOR OF SAFETY FOR GEARS 

The load on the teeth of gears made from forgings may be such as to strain 
the material close to its elastic limit (based upon the worn thickness of tooth), 
if it is free from flaws. For castings this is not a safe rule, as there are always 
hidden defects to a greater or less extent. As long as the strain is kept below 
this point, excessive wear will not take place, but if this point is exceeded but 
slightly, rapid wear, or fracture of the teeth, is sure to result. For reasonable 
service, a factor of safety of 1.5 should be used if the load is uniform. Thus, 
for a forged steel gear having an elastic limit of 20,000 pounds per square inch, 

the safe load would be — = I 3i33° pounds per square inch. For cast 

steel, free from apparent defects, a factor of 2 is recommended; thus, for this 

same strength in steel in a casting the safe load would be = 10,000 

pounds per square inch. 

The elastic limit meant in this connection is the real elastic limit of the ma- 
terial as taken by an accurate extensometer and not by the drop of the beam, or 
by caliper ■ measurement, as has been commercial practice. This instrument 
detects the first indications of permanent set in the test piece, showing that 
the safe load for that material has been exceeded ; the drop of the beam is not 
apparent for some time after this. For untreated mild steels this point is some- 
times at one-half the drop of beam; for the higher grades, however, the two 
points are closer together. 

Such an instrument is described by T. 
O. Lynch in a paper on " The Use of the 
Extensometer for Commercial Work " 
read before the American Society for Test- 
ing Materials ; Philadelphia, published in 
the Proceedings for 1908, volume 8. 

It must be pointed out that gear 
steels should have an ample reduction 
of area to guard against sudden frac- 
ture. Test pieces should be cut with 

the center of tooth a little below the bottom line, say 0.07 of the circular pitch, 
as illustrated by Fig. 50, as it is through this point that the tooth generally 
breaks out. 




FIG. 50. LOCATION OF TEST PIECE. 



66 AMERICAN MACHINIST GEAR BOOK 

The strength of the material in gears will be found to vary as much as 30 
per cent, in different parts of the tooth and rim, therefore a settled point for 
cutting out test pieces is necessary if uniform, safe, or accurate results are to 
be expected. It does not greatly matter if the threaded portion of the test 
piece projects into the tooth space, as it will on all gears 2 X A diametral pitch 
and finer, so long as the 0.505-inch portion of the piece is clear. When a 0.505- 
inch test piece (H of a square inch in area) cannot be obtained in this manner, 
make one 0.2525 inch in diameter (3^o of a square inch in area), leaving the 
threaded portion Y% inch instead of M inch, which is standard. 

Note that elastic limits given in table for wear of gear teeth is by drop of 
beam. 

STRENGTH OF BEVEL GEARS 

In general apply the Lewis formula for spur gears, figuring the safe load 
from the average pitch diameter, or, stated a little differently, the velocity in 
feet per minute and the pitch is to be taken at the average pitch diameter, 
otherwise the gear is to be treated as a spur. 



Let N 


= 


number of teeth. 


P' 


= 


circular pitch. 


D' 


= 


pitch diameter. 


b 


= 


face width. 


P'» 


= 


average circular pitch. 


D & 


= 


average pitch diameter. 


a 


= 


apex distance. 


E 


= 


center angle. 


S 


= 


safe working stress. 


v & 


= 


velocity (average) in feet per minute. 



In order to get the average pitch we must first determine the apex distance a. 
Now 

D' 



a = 



2 sin. E ' 



The average pitch is the pitch at the center of the gear face /. Above this 
section the tooth strength increases and below this point it decreases. The 
mean strength of the tooth is, therefore, located at this point. Thus it is the 
proper dimension to use in determining the strength of the tooth. This average 
pitch is found from the following equation: 



,= >'(»-!). 



a 



SPEEDS AND POWERS 



67 




FIG. 51. DIAGRAM FOR STRENGTH OP BEVEL GEARS. 



This formula is derived as follows: By referring to Fig. 51, it is evident that 
the pitch of the tooth at p' is to the apex distance a as the pitch at p' & is to the 

The average pitch diameter D & is found from formula (3) by substituting the 
average pitch. The average velocity V & is found from formula (4) by using 
the average pitch diameter. Then 

w 



a = 



2 sin. E ' 



-.-'(- t\ 



(2) 



D & = N p' & 0.3183. 

V & = 0.2618 D a (r.p.m.). 

Safe load = S P \by( 6 J°^ v y 



(3) 
(4) 

(5) 



68 AMERICAN MACHINIST GEAR BOOK 

TT Safe load X V & . . 

Horse-power = - — . (6) 

33,000 

The values for the factor y are from the Lewis formula. 

An illustrative example is as follows: What power may be transmitted by a 
pair of miter gears of the following dimensions: 30 teeth, 2-inch pitch, 5-inch 
face, 19.107-inch pitch diameter, at 50 revolutions per minute? The material 
is cast iron. 

19.107 ... .. 

a — — — — = is- Si inches, the apex distance; 

2 X 0.707 ° ° ' F ' 

p' a = — — = 1.63 inches, the average pitch; 

D & = 30 X 1.63 X 0.3183 = 15.5 inches, the average pitch 

diameter; 
V & = 0.2618 X 15.5 X 50 = 203, the average velocity in feet 

per minute; 

The safe load = 8000 X 1.63 X 5 X 0.102 X ( - — 7^ I = 4970 pounds. 

mi , 497° X 203 

I he horse-power = ■ — — — = ^o. c. 

33,000 ^ D . 

The teeth in bevel gears are more strongly shaped than the teeth of spur 
gears of the same pitch and number. This increase is represented by the radius 
e — p' & in Fig. 51, compared with the radius at the point p' . The corresponding 
number of teeth for this larger radius is found by the expression: 

N 
cos. E 

When selecting the constant y, however, it is well to disregard this increase, as 
it will tend to compensate for the loss in efficiency due to the use of bevel gears. 

THE STRENGTH OF SHROUDED GEAR TEETH 

In regard to strength of shrouded gear teeth, Wilfred Lewis submits the fol- 
lowing, originally published in American Machinist of Jan. 30, 1902. 

"I do not know of any careful analyses of or experiments on the strength of 
shrouded gear teeth, but I have some recollection of a tradition in vogue about 
twenty-five years ago that from one-fourth to one-half might be added to the 
strength by shrouding. 

"There are, however, a number of cases to be considered; the shrouding may 
extend to the pitch line only or to the ends of the teeth, and it may be single or 
double. Formerly the practice of shrouding pinions was more common than 
it is to-day, because the advent of steel as a cheap construction material makes 
it possible to obtain unshrouded pinions of greater strength than the cast-iron 



SPEEDS AND POWERS 69 

gears with which they engage, and now steel pinions have generally supplanted 
the old cast-iron shrouded ones which were naturally more roughly shaped, be- 
cause harder to fit, and more difficult to assemble by reason of the shrouding. 
In my investigation of the strength of gear teeth I therefore assumed that the 
time for shrouded gears has passed, at least as far as machine tools were con- 
cerned, but they are possibly used as freely as ever on roll trains and some other 
classes of machinery, so that the problem may still be worthy of consideration 
from a practical standpoint. Rankine, in his 'Applied Mechanics,' rather 
summarily disposes of the strength of gear teeth by assuming the load that may 
be carried on one corner to be all that any tooth is good for. When so loaded, 
it is shown that the corner will break off at an angle of 45 degrees, and the 
strength of a tooth of any width is then no greater than that of a tooth whose 
width is twice its hight. So, if the hight of a tooth is 0.65 pitch, the strength, 
according to Rankine, should be taken for a width of only 1.3 pitch. Faces 
wider than this would be no stronger, and shrouding at one end would make 
no difference. 

" But his assumption is untenable, because no maker of machinery who values 
his reputation will put gears into service bearing only at one end, and should 
they be so started, an even distribution of pressure is sooner or later effected by 
the natural process of wear. 

"A comparison of strength between shrouded and unshrouded gears should 
therefore be made on the assumption of uniform distribution of pressure across 
the faces of their teeth, and for this purpose it will be expedient to neglect the 
influence of tooth forms, which would complicate and prolong the investi- 
gation, and treat all teeth simply as rectangular prisms, which may or may not 
be supported by shrouding. Rankine and Unwin have both been contented to 
estimate the actual strength of teeth as though they were rectangular prisms, 
and, although this is far from the truth, it is certainly more admissible as a 
basis of comparison for another variable than as an approximation for a direct 
result. The effect of this assumption will be to exaggerate the value of shroud- 
ing, and for the present it will be sufficient to indicate roughly the maximum 
benefit to be anticipated. 

"In Fig. 52 a gear tooth is shrouded at one end, and the problem is to de- 
termine its strength as compared with the same tooth not shrouded. For con- 
venience, the thickness, or half the pitch may be taken as unity, and the hight 
as 1 .3. The load W is assumed to be applied uniformly along the end of the tooth 
over the face b, making the full load b W. If there were no shrouding, the 
strength of this tooth would be measured by the transverse resistance at its 
root /, and if broken at the root as shown in Fig. 53, we may consider how much 
strength could be given to it by shrouding alone. 



7o 



AMERICAN MACHINIST GEAR BOOK 



"A tooth thus broken would have some strength as a cantilever imbedded 
in the shrouding, but more as a shaft subjected to torsion, and for the shape 
here assumed the torsional strength alone will probably exceed the combined 
strength due to torsion, and flexure for any actual shape. 

"The rectangular tooth whose sides are h and t cannot be treated as an or- 
dinary shaft because its neutral axis is at one side instead of, as usual, at the 
center of gravity. It must therefore be treated as one half of a shaft whose 



— b- 



m 








liiiB 



-t— J 



-t- 



rn 



FIG. 52. FIG. 53. 

DIAGRAMS ILLUSTRATING THE STRENGTH OF SHROUDED GEAR 

TEETH. 



sides are t X 2 h or 1 X 2.6, for which the moment of resistance is about .6 S, 
where 5 is the shearing stress at the end of a tooth. For the unit load Wwe have 
.6S = 1.3 W, or S = 2.17 W, and for the width b we have-Sj = b S = 2.17 b W. 

"Thus the maximum intensity of shearing S i is found to be a little more than 
twice the full load b W. 

"On the other hand, for an unshrouded tooth the transverse stress/ at the 
root of a tooth depends only upon W and the relation is expressed by the equa- 
tion/ = 7.8 W. In these terms, 

L 
7.8' 



W= -At. 



whence 



/ = 



2.17 b' 
3-5-S-. 



and assuming that the shearing stress S { may be 0.8/, we have b = 2.8. This 
means that a tooth 2.8 wide has as much strength as may possibly be added by 
shrouding at one end, but the question remains to be considered, under what 
conditions and to what extent can this possible strength be made effective? 
The development of stress is always accompanied by strain, and in the case of 
a shrouded tooth the unit load W must be divided between torsion and flexure. 
Obviously, if the tooth is very long, its stiffness under torsion will be so little 
as compared with its stiffness under flexure, that the benefit from shrouding 
will be inappreciable, and on the other hand if very short, the torsional stiffness 
will be preponderate. When a uniform distribution of load has been attained, 
as it must be by the action of wear, that part of a tooth farthest from the shroud- 



SPEEDS AND POWERS 71 

ing will sustain the greatest transverse stress and the load W will be divided at 

all points along the face of the tooth between torsion and bending directly as 

the stiffness encountered in these two directions or inversely as the relative 

strains. Each strain is relieved by the other, but the limit of strength is reached 

when either attains its maximum. 

" Considering a cantilever loaded at the end, we have for the deflection y, 

under the stress/, 

fh 2 

y = — W 

1.5 £ 

where h = 1.3 and E is the modulus of elasticity for flexure. Substituting this 
value of h we have 

1.1/ (1) 

" Considering the tooth as a rectangular shaft in torsion, it will be seen 
that the shearing stress for a distributed load decreases from the maximum S x 
at the shrouding to nothing at the other end of the tooth. For the unit load W 
the shearing stress is S, for an element d x the stress is S d x, and for this stress 
at the distance x from the shrouding, the torsional deflection 

1 O X (JL Jv 

dz = — g- . 

where G is the modulus for shearing. The total deflection for the load dis- 
tributed over a length x therefore is 

Sx 2 

2 G 

or for the face b we have 

Sb 2 

2 G 
" But since G = 0.4 E, we may write 

Sb 2 (2) 

2 0.8 E 
"The value of S has been found to be S = 2.17 W, and we have also found 
/ = 7.8 W, therefore 

* 7 .8 ; ' 

= 0.28/. 

Sb 2 
and 2 = 



may be written 



8E 
o.35 f b2 (3) 



Z= E 



72 AMERICAN MACHINIST GEAR BOOK 

" The deflection of an unshrouded tooth under the load W has been shown by 
equation (i), and, dividing this into equation (3), we have for the relation be- 

tween y and z — = 0.36 2 . 

For a very narrow tooth, letting b = 1, we have z— 0.33;; but since y and z are 
necessarily equal, when the tooth under consideration is attached at its root 
and also to the shrouding, the load W will be supported at both points, and it 
will necessarily be divided between them in the proportion of y to z, or as 1 to 

W 

3. The shrouding will carry — , or 0.77 W, and the root of the tooth 0.23 W. 

" Similarly making, b = 2, or one pitch, we have z = 1.2 y, and at this point 
the shrouding will carry 0.45 W and the root of the tooth 0.55 W. Also, when b = 
3, we have z = 2.7 y, reducing the load on shrouding to 0.27 y and increasing the 
load at the root to 0.73 W. At b = 4, or 2 pitch, z = 4.8 y, the load on shrouding 
drops to 0.17 W and the load at the root rises to 0.83 W. 

"The average width of gear faces is probably about 2.5 pitch, and for b = 5 
we have about 0.12 W carried by the shrouding and 0.88 W carried by the tooth 
acting as a cantilever. We may therefore conclude that the strength of an 
ordinary pinion shrouded at one end only is not increased more than 1 2-88 or 
about 13 per cent., by the shrouding. Indeed, this is only the result of a first 
approximation, and for the successive proportions of W thus credited to the 
shrouding new values of z might be" estimated to be used as the basis of a second 
approximation. But we will not continue the process — it is sufficient to know 
that our valuation of the effect of shrouding is high. A double-shrouded 
pinion running with a gear whose face is 2.5 p f will be about 3 pitch between 
shroudings and its strength will be about the same as that just found for b = 3. 
An ordinary pinion will not therefore be increased in strength by double shroud- 
ing more than 37 per cent., and it is probably safe to say that a more elaborate 
investigation will reduce the additional strength to 10 per cent, for single 
shrouding and 30 per cent, for double shrouding. 

"When the shrouding extends to the pitch line only, the shearing strength of 
its attachment to a tooth is reduced, but the elastic relations upon which the 
strength at the root depends remain practically the same. In this case the 
shearing strength instead of the transverse strength limits the strength of a 
tooth, and the strength is apparently less than for full shrouding. 

" The effect of shrouding is clearly to prevent the adjacent part of a tooth from 
exercising its strength as a cantilever. The shrouding carries what the tooth 
itself might carry almost as well. A heavy link in a light chain adds nothing to 
the strength of the chain, and teeth which are not strong all over need not be 
strengthened in spots. A little more face covering the space occupied by 



SPEEDS AND POWERS 



73 



shrouding is more to the purpose for durability as well as for strength, and when 
this fact is appreciated I believe the practice of shrouding will disappear in 
rolling mills, as it has done in machine shops. 

"In regard to the working stress allowable for cast iron and steel, I may say 
that 8000 pounds was given as safe for cast-iron teeth, either cut or cast, and 
that 20,000 pounds was intended for ordinary steel whether cast or forged. 
These were the unit stresses recommended for static loads, and as the speed 
increased they were reduced by an arbitrary factor, depending upon the 
speed. 

"The iron should be of good quality capable of sustaining about a ton on a 
test bar 1 inch square between supports 1 2 inches apart, and of course the steel 
should be solid and of good quality. The value given for steel was intended to 
include the lower grades, but when the quality is known to be high, correspond- 
ingly higher values may be assigned. 

"In conclusion I may say that the crude investigation here given seems to 
justify the traditions referred to that from M to M may be added to the 
strength of teeth by shrouding. If the teeth are very narrow, Y A may be added, 
but generally, I believe, M is enough and since writing the above I find that 
D. K. Clark almost splits the difference by adding Yz for double shrouding. 
But the development of the full strength of gear teeth depends nearly as much 
upon the strength and stiffness of the gear journals as upon the teeth them- 
selves, and no rules can be given for indiscriminate use." 



WEAR OF GEAR TEETH 

The Lewis formula is the only accurate method of figuring the power of gears so 
far as the strength of the teeth is concerned, but takes no account whatever of 
wear, and the value of the tooth surfaces to resist crushing of the material. 
Trouble from this source is a common experience, although not properly under- 
stood, as it is sometimes difficult to account for the "mysterious" failure of 
gears that were apparently of ample strength. It is noticed that gears gen- 
erally fail through wear and not by fracture of the teeth, also that the teeth 
often break at a load far below that which is considered safe. More atten- 
tion, therefore, should be given to this point. 

A certain combination of diameters will carry but a certain load per unit of 
face, irrespective of the pitch of the gears, so that there is no gain in an in- 
creased pitch above that just sufficient to resist fracture. This pitch may be 
found as usual by the Lewis formula, but the actual strength of the material 
should be used. The material in a i-inch pitch tooth is stronger proportion- 



74 AMERICAN MACHINIST GEAR BOOK 

ately than the same material in a 2-inch pitch tooth. This is caused by the fact 
that nearer the exterior the material is stronger and of a closer grain, due to 
rapid cooling. A tooth is stronger at the top than at its root. It would seem 
as if tests of material should be made for flexure and not for tensile strength, as 
the tooth breaks through bending. The average ultimate strength of cast iron 
for flexure is 38,000 pounds per square inch, while the tensile strength is 
24,000 pounds per square inch. 

Aside from this feature the surface hardness should also be consid- 
ered irrespective of the strength of material. For instance, a pressure of 5000 
pounds per unit of contact would be allowed for a case-hardened steel surface, 
while but 1500 would be allowed for the same material in its unhardened 
condition. 

The relative hardness of material, in conjunction with the co-efficient 
of friction for different grades and hardness of material engaging will supply 
the safe load A per unit of area. 

It is true that the arc of rolling contact in gears is very small ; the balance is 
sliding contact, which increases proportionately over the rolling contact as the 
pitch points separate, or as the tooth disengages, and decreases as the tooth en- 
ters contact until the pitch points again engage where it is rolling. 

The wearing qualities of the teeth depend greatly upon their condition 
when put into service. If a little care is used to obtain a smooth surface 
at the start and allow the teeth to find their proper bearing, the gears will wear 
indefinitely longer than if put under full load when new, no matter how 
accurately the teeth are cut. Also a gear once started to cut can often be 
saved by the timely application of a fine file, finally smoothing the teeth with 
an oil stone. 

A series of experiments to determine the proper load per unit of contact (A ) 
for different grades and hardness of material used would certainly lead to a 
fuller knowledge of the capacity of gears for the transmission of power and leave 
less to supposition on the part of the designer. On account of the peculiar 
nature of the tooth contact it is quite likely that the best manner to reach ac- 
curate results would be with gears made from the materials under considera- 
tion. The values given in Chart 2 are the best obtainable at the present 
writing. 

The idea of limiting the load to the proportion of the gear diameters ir- 
respective of the pitch (for a unit of face) may at first appear startling, but 
when we consider that the radii from which the tooth is drawn are always 
proportional to the pitch diameter of the gear and not to the pitch, and that the 
teeth in contact are actually two cylinders rolling and slipping upon each other, 
it appears more reasonable. See Fig. 54. It should be understood, however, 



SPEEDS AND POWERS 



75 



that the diameter and position of these rollers change constantly throughout 
the contact, and that a gear made in strict accordance with Fig. 54 would not 
give a uniform movement. It illustrates the principle however. 





FIG. 54. GEAR TOOTH ACTION. 



FIG. 55. CURVATURES. 



To secure safe results the flank or shortest radius is used in these formulas. 

The following is the gist of an article by Harvey D. Williams, in the American 
Machinist, with its application to gear transmissions: 

"The curvature of a plane curve is defined by mathematicians as the change 
of direction per unit of length, and is equal to the reciprocal of the radius of 
curvature at the point considered. Thus in going once around a circle of radius 
R the distance traversed is the circumference 2 tt R, while the change of direc- 
tion is in circular measure or radius 2 v. 

"The change in direction per unit of length is therefore 



2 7T 



2 7T R 



R 



"Accordingly the curvature of a 2 -inch circle is 1, that of a i-inch circle is 2, 

that of a X A -inch circle is 4, and that of a straight line is - - = o, etc 

00 

" The curvature of a straight line being zero, that of an arc may be said to be 
its curvature in relation to the straight line, or its relative curvature to the 
straight line. Similarly, in comparing the curvature of two arcs, it will be con- 
venient to use the term ' relative curvature ' instead of the difference of curva- 
ture, meaning thereby the algebraic difference of the curvature as dimensioned 
in Fig. 55. 



7 6 



AMERICAN MACHINIST GEAR BOOK 



"It will be seen that when two plane curves are tangent to each other ex- 
ternally the relative curvature is to be found by adding the respective curva- 
tures, and when they are tangent internally the relative curvature is to be 
found by subtracting. . . . 

"The amount of contact between plane curved profiles is measured by the 
reciprocal of the relative curvature." See formulas 21, 22, and 23. 




pig. 56. 



fig. 57. 

CONTACTS OF THE SAME CURVATURE. 




In each of the cases shown by Figs. 56, 57, and 58 the contact is 4, and if these 
profiles were made of the same material and the same width of face, they would 
be equally efficient as regards their ability to withstand pressure. 



Let 
C 
R 1 

r 1 

f 
V 

A 

D 1 

d> 

a 

W c 

W 



DEVELOPMENT OF FORMULAS 

contact. 

flank radius of the gear. 

flank radius of the pinion. 

face width. 

velocity in feet per minute. 

safe crushing load per unit of contact. 

pitch diameter of the gear. 

pitch diameter of the pinion. 

angle of obliquity. 

safe load on the tooth to resist crushing and wear, 

safe load on the tooth to resist fracture. 



SPEEDS AND POWERS 



77 



Then 
and 

For spur gears 

For internal gears 

For racks 



D 1 

R l = ■ sin a, (19) 

d 1 



sin a. (20) 

C = (21) 

C = — -^-— - (22) 



R 1 ' 



I Y 

C = = — - = r\ (23) 



± o 



r 



w ' = c '! A (w)' ^ 



It is desirable that the gear and pinion should wear equally to avoid the 
necessity of engaging a new pinion with a partly worn gear, thereby decreasing 
the life of both. It is assumed that wear is proportional to the hardness of the 
material ; obviously the pinion should be harder than the gear in proportion to 
the ratio of the drive. Therefore, to secure equal wear in a pair of gears having, 
say, a ratio of 4 to 1, the pinion should be made four times as hard as the 
gear. 

I have thought that a hard pinion would tend to preserve a softer gear, but 
as no data are found to sustain this theory, and as the value calculated for 
Chart 1 tends toward a much softer gear than was originally thought proper to 
make the best wearing combination for certain ratios, this has not been taken 
into account. It is assumed, therefore, that a hard pinion will neither preserve 
nor influence the wear of a softer gear. Therefore, it may be assumed safely 
that a hard gear will not influence the wear of a softer pinion. Chart 1 is made 
on this basis. The wear of gear and pinion are determined independently if the 
proper combinations of hardness are not used. 

The wear is based entirely on the pinion hardness, the gear performing the 
same amount of work, but having less wear on account of the greater number 
of teeth in use in proportion to the ratio of the gears. For instance, in a gear 
drive having a ratio of 4 to 1, the gear may be 75 per cent, softer than the pinion 
for equal wear. 



78 



AMERICAN MACHINIST GEAR BOOK 



Thus the wear of the gear is found according to the pinion hardness that is 
proper for the ratio of the gears irrespective of the material actually used for 
the pinion. 



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Hardness of the Pinion 



For example: If a pinion of a hardness represented by 0.15 (see Chart 1) 
engages a gear of 0.35 hardness, the ratio being 4 to 1, the wear of the gear will 
be in accordance with the pinion hardness found opposite the line of ratio and 



SPEEDS AND POWERS 



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8o 



AMERICAN MACHINIST GEAR BOOK 



over the gear hardness (0.35), which in this case is 0.173^. In this event the 
gear will wear out first. On the other hand, if the pinion had been of 0.20 hard- 
ness, the pinion would wear out first. The value for the wear of the gear would 
remain o.i^ 1 ^ 





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Pitch Diameter of the Pinion in Inches 



It is thought that the elastic limit of a material follows the hardness, there- 
fore the wear may be determined from the elastic limit. The points of hard- 
ness, however, are used in the accompanying charts for convenience; Chart 2 



SPEEDS AND POWERS 8 1 

gives the corresponding values. The hardness values given in these tables were 
obtained by pressing a K-inch hardened steel ball into the surface of the ma- 
terial with a pressure of 10,000 pounds. The dimensions given are diameters 
of the indentations thus made. See Fig. 59. The comparisons in hardness were 

h -H -I 

I i 



7/777777'/ 
FIG. 59. CHORD MEASUREMENT OF HARDNESS TEST. 

made inversely to the square of these diameters. The elastic limit was de- 
termined for one of these values; the comparison for others may be found by 
the same inverse proportion. 

Thus, the square of 0.20 = 0.04 and the square of 0.30 = 0.09. Therefore, 

0.20 would be — - — = 2M times harder than 0.30. This is a proper combina- 
tion for a gear ratio of 2^ to 1. The elastic limit for 0.20 is 60,000 pounds per 
square inch. The elastic limit of 0.30 = ■ — ''—. — = 26,700 pounds per square 

2/4 

inch. 

For comparison with the Brmell scale; hardness value 0.22 inch in Chart 2 
measures 4.6 millimeters; the hardness numeral for this impression being 167, 
the impression is made with a 10-millimeter ball at a pressure of 3000 kilo- 
grams. 

As the elastic limit of cast iron is very close to its ultimate tensile strength 
the ultimate strength may be used to determine the hardness. This was at first 
very confusing before it was found that the hardness followed the elastic limit, 
as cast iron under the ball test referred to would show a hardness equal to ma- 
chine steel of twice its ultimate tensile strength. 

The values of A according to the hardness or elastic limit of the material 
(Chart 2) have been assumed as correct for gears operating 10 hours per day 
for a period of two years. If found in error their multiplier given in Table 
15 for the time and conditions of service may be shifted without changing the 
original values of Charts 1 and 2. This table may be elaborated to any de- 
sired extent to cover various conditions. It is evident that a pair of gears 
will not last as long fastened to the ceiling or to insecure timbers as if mounted 
upon a proper concrete foundation. There are all manner of machine construc- 
tions to be considered as well as unknown overloads, the influence of fly-wheels 
and other things that are usually neglected. All this, however, is simple in- 
deed when the Stygian darkness in which we are now wandering is considered. 



82 



AMERICAN MACHINIST GEAR BOOK 



TIME OF SERVICE 



For 3 months 
For 6 months 
For 9 months 
For i year. . . 
For 2 years. . 
For 3 years . . 
For 4 years . . 
For 5 years. . 
For 6 years . . 
For 7 years . . 
For 8 years . . 
For 9 years . . 
For io years. 



UNIFORM LOAD 



CONTINUOUS 



3.OO 
I.50 
I. OO 

0.75 
O.38 
O.25 
O.I9 

O.I5 
O.I3 
O.II 
O.IO 

0.09 

0.08 



IO HOURS DAILY 



8.00 
4.OO 
2.67 
2.00 
I. OO 
O.67 
O.50 
O.40 

o-33 
0.29 
0.25 
0.22 
0.20 



5 HOURS DAILY 



18.OO 
9.OO 
6.00 

4-SO 
2.25 
I.SO 

i-i3 
0.90 

0.75 

0.65 
0.56 
0.50 
o.45 



Table 15 — Multipliers for Factor 'A' 
According to the Conditions of Service and Desired Life of Gears 

For gears subjected to 25 per cent, overload, multiply result by 0.80; for 
gears subjected to 50 per cent, overload, multiply result by 0.70; for gears 
subjected to 75 per cent, overload, multiply result by 0.60; for gears subjected 
to 100 per cent, overload, multiply result by 0.50; for gears operating in dust- 
proof oil case, multiply result by 1.50. 



EXAMPLES 

What is the safe load for a pair of spur gears properly mounted on concrete 
foundations to operate continuously for a period of five years before replacing? 
The gears are to run in oil in a dust-proof case driving an electric generator 
making 300 revolutions per minute from a turbine revolving at 1200 revolutions 
per minute. The overload at no time will exceed 25 per cent. The gear has 
84 teeth, 3 diametral pitch, 12-inch face and 28-inch pitch diameter. The 
pinion has 21 teeth, 3 diametral pitch, 12-inch face and 7-inch pitch diameter. 
The ratio is 4 to 1. The circumferential speed at the pitch line is 2200 feet per 
minute. The pinion should be four times as hard as the gear for equal wear 
and, according to Chart 1, if a cast-steel gear of 30,500 pounds' elastic limit 
which is assumed, to have a hardness value of 0.28, the pinion hardness should 
be represented by 0.14, which represents an elastic limit of 120,000 pounds 
per square inch. This may be obtained by hardening, chrome nickel or other 
high-grade steel. High-carbon steel should be avoided for this purpose on 
account of its tendency to crystallize. 



SPEEDS AND POWERS 



83 



Referring to Chart 2 it is found that the temporary value of A is 7500 pounds. 
The multiplier for time of service is 0.15 ; the multiplier for overload is 0.80; the 
multiplier for the oil case is 1.50. Thus the final value of A = 7500 X 0.15 X 
0.80 X 1.50 = 1350 pounds. 

In the formula 

= CfA 6 °° 



where 



then 



or 



W c 

W c 
C 

f 

A 
W c 



600 + V ' 

= the safe load in pounds (to be determined), 
= 0.70 (from Chart 3), 
= the face, 12 inches, 

= 1 3 so, and ; . (from Chart 4) = 0.21, 

OJ • 600 + 2200 v 

= 0.70 X 12 X 1350 X 0.21 = 2380 pounds, 

2380 X 2200 



33,000 



= 158 horse-power. 



0.10 


































































































































































0.20 


















































































































































































































































>, v.ou 


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t> 

+ 0.50 



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90 


































































































































































1.00 



















































































200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000 

Velocity in Feet per Minute 



CHART 4. RELATIONS BETWEEN THE QUANTITY 



600 



600 + velocity 



-AND THE VELOCITY. 



The strength of the teeth must now be checked by the Lewis formula to 
guard against fracture at this load. We find by this method a safe working 
load of 317 horse-power. 

This illustrates that the teeth are capable of carrying 317 horse-power, but 
as shown by the value W c they would wear out in about one-half the specified 
time if such a load were applied. If the example had read " ten hours per day" 



84 AMERICAN MACHINIST GEAR BOOK 

instead of "continuous," all other conditions remaining the same, we would 
have: A = 7500 X 0.40 X 0.80 X 1.50 = 3600 pounds, 

3600 X 2200 

■ = 240 horse-power. 

33,000 ^ 

For this load, however, the teeth would be liable to fracture. 

RELATIVE IMPORTANCE OF STRENGTH AND HARDNESS 

It would appear that the actual strength of the tooth is to be a secondary 
consideration, figured only as a preventive against fracture. The real points 
to be considered are: First, the proper proportion of the gear diameters; second, 
the hardness of the material and the best combination of hardness for wear. 
George B. Grant was very near the truth in saying, "It does not proportionately 
increase the strength of a tooth to double its pitch. " With herringbone gears 
it would seem that the strength need hardly be considered, as it is practically 
impossible to break out a single tooth of sufficient angle, an entire section 
must be removed. They must be worn out. 

Aside from the hardness, the value of the material to avoid crystallization 
must be considered, as gears in which the teeth are apparently extremely tough 
will often become brittle and drop off after a comparatively short service from 
this cause. For this reason high-carbon steel should be avoided and hardness 
obtained by case-hardening, or by the addition of manganese, nickel, chromium, 
vanadium, or some other, hardening ingredient. 

It is hardly necessary to add that proper lubrication adds greatly to the ef- 
ficiency of a gear drive, except, of course, where they are exposed to brick or 
cement dust, where it is often advisable to run them dry, as the oil will hold 
particles of grit and cause the teeth to cut. A jet of air applied at the point of 
contact is also found beneficial in such cases, as it will remove particles of grit 
from the teeth before they enter contact. 

The constructions of housings upon which the gears are mounted is of the 
utmost importance, as the absence of vibration is essential to high efficiency. 
Where the housings are insecure it is often found that a rawhide pinion will 
sometimes give better service than one made of iron, as the rawhide will give 
and absorb vibration that would destroy the harder material. 

Another point that naturally suggests itself is the proper value of a suitable 
lubricant. It is evident that the efficiency of 95 per cent, running dry and 98 
per cent, when immersed in oil does represent the total saving. Lubrication 
means in many cases the difference between a successful drive and a failure 
which is not apparent from any superficial tests made for power efficiency only. 

The question is often asked, " What is the life of a gear? " It is evident that 



SPEEDS AND POWERS 



85 



continual service will wear out any material no matter how hard. We will say 
for example that a load of 900 pounds per unit of area will wear out a pair of 
cast-iron gears in one year's continual service. What pressure must be applied, 
therefore, to wear out these gears in six months or to allow them to run for five 
years? It would appear that this could be taken care of nicely by factor A, 
and as in Table 15, provided, of course, that other conditions are correct and 
that we have the proper analysis of our materials. We should determine not 

only what grades of material wili 
wear best together, but also how long 
they will wear. 

A gear may be said to be worn out 
when the teeth have been reduced to 
not less than one-half their original 
thickness — this will subject them to 





fig. 60. 



INTERNAL GEAR MESHING WITH 
SPUR. 



FIG. 6l. 



LIMITING WEAR OF GEAR 
TEETH. 



the limit of ultimate stress of the material if allowed their full load according 
to the Lewis formula. If this is exceeded the teeth would be liable to fracture. 
See Fig. 61. 

It is evident that during the latter part of the life of a gear the teeth will wear 
more rapidly, as the backlash will allow the teeth to hammer with variations 
in the load or in reversing. 

It is well known that all materials are subject to what is known as fatigue, 
that is, a piece of steel that will stand an intermittent strain of 2000 pounds 
successfully is liable to fail if this load is permanently applied. This should 
enter into our problem, as it is often required to design a pair of gears to trans- 
mit a certain load continuously, with the guarantee that they will render suc- 
cessful service for, say, two years before renewing. The correct solution of this 
problem would require a thorough knowledge on all the points brought up in this 
paper, also a proper determination of their efficiency, unless, of course, the 
gears were made amply large, or there had been some precedent upon which 
to base the calculations. 

Another point to be considered is what difference (if any) should be made in 



86 AMERICAN MACHINIST GEAR BOOK 

the comparative hardness in favor of the pinion, so that it may have wear 
proportional to the ratio of the pair. 

IMPORTANCE OF PROPER DESIGN 

Aside from all these conditions, to obtain anything like accurate results the 
gears must be mounted in such a manner as to obviate practically all vibration. 
The teeth must be accurately formed and spaced to insure that the impulse 
received by the driven gear is uniform and without variation, and the thickness 
of the teeth must be such as to avoid practically all backlash, except just enough 
to secure free operation, as teeth are often broken by crowding on close centers. 
The gears must also be properly designed to withstand any strains to which the 
teeth are subjected. The proper distribution of the material will add greatly 
to the wear and strength. It is well to have the gear as rigid as possible. It 
should be remembered that accuracy in cutting the teeth will be of little avail if 
they are not correctly mounted upon their shafts. If the shaft is a little under 
size, the key will cause it to run out of true. This will also apply when the shaft 
is a neat fit and the taper key is driven too tightly. 

In the absence of practically all experimental data I have not attempted to 
put forward anything more than a general outline of the situation and to bring 
up essential points for consideration. It is trusted that they will be received as 
such. In view of the growing importance of gears for the transmission of power 
the points referred to are certainly worthy of attention. 

SPEED OF SPUR GEARS 

When the question is asked, "What is the greatest circumferential speed at 
which a spur gear may operate?" we are told, "When 1200 feet per minute is 
exceeded either rawhide or herringbone gears must be used, and even when 
properly mounted 3000 feet is the limit of speed for any gear." 

To illustrate the fallacy of this statement consider a pair of spur pinions, each 
of 12 teeth, 6 pitch, 2-inch pitch diameter, running at a speed of 1200 feet per 
minute. A moment's consideration will show that the noise generated by such 
a drive would be excessive, as this would mean 2280 revolutions per minute. 
These gears would make their presence known at a speed of 400 feet per minute, 
which represents 761 revolutions per minute. 

At first thought it appears that the number of teeth in contact would rep- 
resent the comparative speed value, but, as it is sometimes possible to obtain 
as many teeth in contact in a small pair of gears as in a large pair, owing to the 
difference in pitch, the proportion of gear diameter must also be taken into 
account. As the proportionate value of the gear diameter is represented by the 



SPEEDS AND POWERS 



87 



number of teeth in contact, the pitch remaining constant; this value may be 
gaged according to the circular pitch. Fig. 62 will illustrate this. 

The relative speed value is found in 
the product of the number of teeth in 
contact and the circular pitch, or n p' . 

n = number of teeth in contact, 
p' = circular pitch. 

The next step is the determination of 
a factor from actual practice, p, which 
multiplied by the product of the pitch, 
p f , and the number of teeth in contact, 
n, will give the safe speed. The formula 
now becomes: 

Safe speed = p' n p. (26) 

The factor p may be made to cover al- 
most any type of gear or condition of 
service. If the safe speed is known for 
certain machine construction, the value 

of the factor p may be determined as below for similar drives. 

Safe speed 




FIG. 62. DIAGRAM ILLUSTRATING METHOD 
OF DETERMINING NUMBER OF TEETH 
IN CONTACT. 



Value of p = 



p n 



(27) 



For general use the values of p have been estimated as in Table 16. 



STYLE OF GEAR 



Spur gear, pattern molded 

Spur gear, machine molded 

Spur gear, commercial cut 

Spur gears, cut with exact cutters, accurately spaced 

Spur gears, cut stepped teeth 

Spur gears, fiber 

Spur gears, rawhide 

Herringbone gears, angle of spiral, 10 degrees 

Herringbone gears, angle of spiral, 20 degrees 

Herringbone gears, angle of spiral, 30 degrees 

Herringbone gears, angle of spiral, 45 degrees 



COMMERCIAL 


GENERATED 


CUT GEARS 


TEETH 


to 300 




no to 450 




600 




700 


80O 


820 




900 


IOOO 


IOOO 




700 


1 100 


800 


1400 


I IOO 


1900 


2400 


4000 



Table 16 — Values of Factor /> 



88 AMERICAN MACHINIST GEAR BOOK 

NUMBER OF TEETH IN CONTACT 

As the necessary formula to determine the number of teeth in contact is 
considered too cumbersome for practical use, a graphical solution is given, as 
follows : 

Referring to Fig. 62, the length of contact is measured between the intersec- 
tion points of the line of pressure and the addendum circles of the gear and 
pinion, or between a and b. In case this intersection falls outside the inter- 
section of a line drawn at right angles with the pressure line to the center of 
the gear as at the point c, the contact is measured from the point c, as this 
indicates that the distance a c must be deducted for interference. The gear 
tooth is rounded from this point out, or the flank of the pinion tooth is under- 
cut to accomplish the same purpose. 

If, on the other hand, this point falls outside the intersection of the pressure 
line and the addendum circle as at c' this extra length must be deducted, as 
there is no contact until the point of the mating tooth has passed the addendum 
circle. In order that the number of teeth in contact can be stepped off, the lines 
0' c and b are extended to the pitch or to points e and/. The length of con- 
tact is then measured from e to d on the gear, and from d to / on the pinion 
along this pitch circumference. This distance divided by the circular pitch 
equals the number of teeth in contact. 

LIMITING SPEEDS 

Estimate the maximum speed to avoid danger of fracture for the best type of 
gear and condition of service as 500 feet per minute per 1000 pounds safe 
working stress of the material of which the gear is constructed. Thus the 
maximum speeds would be as follows : 

4,000 feet per minute for cast iron of 8,000 pounds per square inch. 
8,000 feet per minute for cast steel of 16,000 pounds per square inch. 
10,000 feet per minute for machinery steel of 25,000 pounds per square inch. 

To attain anything like these speeds, however, the gears must be exception- 
ally accurate and well balanced, also the housings must be sufficiently heavy 
to obviate practically all vibration. The restriction placed by the above 
limits, however, will avoid the possibility of allowing the higher speeds. 
According to a series of experiments made by Prof. Charles H. Benjamin, 
American Machinist, December 28, 1901, page 142 1, the bursting speed of a 
solid, cast-iron gear blank is found to be 24,000 feet per minute. The centrif- 
ugal tension at this speed is 15,600 pounds per square inch. The same wheel, 
split between the arms, burst at an average speed of 11,500 feet per minute. 



SPEEDS AND POWERS 89 

CONSTRUCTION OF THE GEAR 

Approximate multipliers should be used for various designs in reference to 
limiting speeds. 

Properly proportioned solid gear '. 1 . 00 

Gear split through the arms o. 75 

Gear split between the .arms o. 50 

Link flywheel construction , o . 60 

These values correspond closely to those given by the Fidelity and Casualty 
Company, which sets the limit of speed for a solid flywheel (cast iron) at 
6000 feet per minute. No value is given for the wheel split through the arms. 
The principal cause of failure (as pointed out by Professor Benjamin in the 
article above referred to) in gears split between the arms is the necessity of 
placing bolting lugs on the inside of the rim. These lugs naturally tend to 
increase the stress at their point of location and fracture the rim in this locality 
at correspondingly low speeds. 

LOCATION OF THE GEAR 

The influence of the location on the speed cannot be well determined. In 
general, however, for gears mounted upon insecure foundations or secured to 
the wall or ceiling of light buildings more or less allowance must be made. 
One case in mind is a 10-horse-power motor direct-connected to a line shaft 
making 150 revolutions per minute. The motor is securely bolted to the 
joists: the gears have 106 and 22 teeth, respectively, 4 diametral pitch, 5-inch 
face, cast iron and rawhide; the speed is 1050 feet per minute. These gears 
were exceptionally noisy and were replaced by herringbone gears of the same 
normal pitch, with the angle of spirality 20 degrees. The gear in this instance 
was cast iron and the pinion machinery steel. These gears were no better 
than the first pair and were replaced by gears with a spiral angle of 45 degrees. 
These proved to be but little better, and as a last resort a pair was installed 
with an angle of 76 degrees. These gears were fairly quiet when operating 
under full load, but very distressing when running light. This condition is 
sometimes found in spur gears with excessive backlash. It was noticed that 
the teeth gave evidence of rapid wear due to the reduced face contact. Rubber 
cushions placed under the motor and shaft hangers and filling the space be- 
tween the hub and rim of gear with wood made no perceptible difference in 
the noise of this drive, which was finally abolished. 

In this connection I have knowledge of a pair of herringbone gears, cast 
iron and machine steel, 80 and 24 teeth, 6 normal pitch, 20- and 6-inch pitch 



90 AMERICAN MACHINIST GEAR BOOK 

diameters, angle of spiral 45 degrees that are practically noiseless at a speed of 
3150 feet per minute. At this speed, however, the load is comparatively light. 
These gears are driven by a 10-horse-power motor and are entirely satisfactory. 

RELATION OF PRESSURE TO SPEED 

The question naturally arises, "In what way does the tooth pressure in- 
fluence the speed of gears? " From a power standpoint this is ordinarily taken 

care of by Mr. Barth's expression - — -. But this does not fix the speed at 

600 + V 

which noise may be avoided, which is the subject of this paper. 

It has been assumed that the tooth pressure allowable at the speeds given 
by the formula (26) are within the limits placed by the foregoing formula (24) 
for wear. 

As previously pointed out "the strength of teeth in herringbone gears need 
hardly be considered, as it is impossible to break out a single tooth provided 
the angle be great enough to engage another tooth before the first lets go." 
However, the actual contact, which depends upon the angle, should be used 
instead of the actual face when determining the load for wear, as increasing 
the angle decreases the noise, but increases the wear. 

ILLUSTRATIVE EXAMPLE 

Required the safe speed of a pair of solid steel spur gears of the following 
dimensions: 

Gear, 80 teeth, 2-inch pitch, 6-inch face, 50.93-inch pitch diameter, pinion 
48 teeth, 2-inch pitch, 6-inch face, 30.558-inch pitch diameter. 

Cutters for these gears to be made for the exact number of teeth. 

The computations for speed involve the number of teeth in contact =2.5, 
value of p according to Table 16 = 800, safe speed = p' n p = 2 X 2^2 X 800 
= 4000 feet per minute. According to factors given for limiting speeds, these 
gears would be amply safe to resist fracture, but would be at the extreme 
limit for cast iron of a safe stress of 8000 pounds per square inch. The com- 
putations for wear involve, assuming that the pinion is made from steel of 
an elastic limit of 50,000 pounds per square inch, the limiting tooth pressure 
at this speed would be equal to the value W c as follows: 

W c = Cf A , T . = 2.4 X 6 X 3000 X 0.13 = 5600 pounds, 

Ooo + V 

in which 

C = 2.4, 

/=6, 

A = 3000, 



SPEEDS AND POWERS 9 1 

and 

600 

600 +v = °- 13 - 

The computations for strength involve the use of the Lewis formula, based 
on carrying the entire load on one tooth, 

W = S p' f y — — q— T7 = 25,000 X2X6X0.111X 0.1.3 = 43 2 9 pounds, 

in which 

5* = 25,000 pounds per square inch (one-half the elastic limit), 

p f = 2 inches, 

/ = 6 inches, 
and 

y = o. 1 1 1 for 48 teeth. 

As the load should not exceed the strength of the tooth, this pressure is the 
limit to be used. The corresponding horse-power transmitted would equal 

W V 4329 X 4000 
■ = - = c;24 horse-power. 

33P°° 33>°°° 

These gears if properly mounted should be satisfactory at a speed of 4000 
feet per minute, the value W c indicating that they could be used for 10 hours 
per day service for two and one-quarter years at a uniform pressure of 4600 
pounds. If longer life or service per day is required the load on the teeth 
must be reduced accordingly. 

HIGH SPEED GEARING* 

For the transmission of power it frequently becomes necessary to use toothed 
gearing, subjected to high peripheral speed conjointly with high pressure per 
unit of tooth contact, and the object of these remarks is to record what has been 
successfully done in recent years, as much higher speeds are now successfully 
attained than formerly. Considered in a static sense, the gear tooth satisfies 
the condition of stress if it is proportioned to endure forces acting transversely 
on it, and the pressure per unit of contract is not of such intensity as perma- 
nently to deform the curved bearing surface of the teeth. When in motion, 
the curved surfaces slide upon each other as they enter and leave contact, and 
when this sliding action is accompanied with high pressure, the limit of en- 
durance is soon reached, and in the case of the inferior materials this occurs 
at comparatively low speeds and pressures. In addition to this, more or less 
impact usually occurs, especially when the resistance is of a fluctuating char- 
acter or the loads are suddenly applied. The effects of this hammering action 

* A paper read before the Engineers' Club of Philadelphia, by James Christie. 



92 AMERICAN MACHINIST GEAR BOOK 

are discernible by a flattening of the curved faces of the teeth, after which the 
proper engagement of the teeth ceases and the gear is speedily destroyed. 

To prevent this, it is desirable to cut the teeth so accurately that no side 
clearance or "backlash" exists, and this is now usually done on first-class 
gearing of even the largest dimensions. Owing to the low elastic limit of cast 
iron and the bronzes we cannot expect these metals to endure so high a pressure 
as steel, and steel appears to be the most trustworthy material to endure the 
highest pressures and speeds. This assertion, however, does not apply to 
all grades of steel. Soft steel surfaces abrade or cut very readily despite all 
methods of lubrication, and surfaces of this material should never be allowed 
to engage in sliding contact. Gearing of soft steel is usually destroyed by 
abrasion at quite moderate speeds. Rolling-mill pinions of steel, containing 
0.3 per cent, carbon, have been destroyed in a few months, whereas the same 
pattern in steel of 0.6 per cent, carbon has done similar work for several years 
without distress. Of course it is necessary to shape the teeth to a proper 
curve to insure proper engagement and uniform angular velocity. 

Some years ago there was required suitable gearing to connect the engines 
to a rolling mill in this vicinity. The diameters of the wheels were 37.6 and 
56.4 inches respectively. They were intended to revolve at speeds of 150 
and 100 revolutions per minute and expected to transmit about 2500 horse- 
power. The character of the service was such that renewal was a serious 
matter and long endurance very desirable. A high grade of steel was selected 
especially in the pinion, in which the greatest wear would occur, and which, 
owing to the location, was the most difficult to replace. The pinion was forged 
from fluid compressed steel of the following composition: 

Per cent. 

Carbon o . 86 

Manganese 051 

Silicon o. 27 

Phosphorus and sulphur, both below o. 03 

The spur wheel was an annealed steel casting: 

Per cent. 

Carbon o . 47 

Manganese o . 66 

Phosphorus and sulphur, both o. 05 

The tooth dimensions were: Pitch, 4.92 inches; face, 24 inches. See Fig. 

63- 

These were accurately cut with involute curves generated by a rolling 
tangent of 16 degrees obliquity. No side clearance was allowed. After 



SPEEDS AND POWERS 



93 



starting the mill, it was found that a higher speed was practicable than was 
originally contemplated. Higher pressures on the teeth were also applied, 




Fig. 65 



EXAMPLES OF HIGH SPEED TOOTH GEARING. 



so that ultimately about 3300 horse-power was transmitted through the gear- 
ing, corresponding to a pressure of nearly 2100 pounds per inch of face. The 
speed was variable, but occasionally attained a velocity of 260 revolutions per 



94 AMERICAN MACHINIST GEAR BOOK 

minute for the pinion, corresponding to a peripheral velocity of 2500 feet per 
minute. This gearing has been in constant operation for several years and 
behaves satisfactorily. 

The highest recorded speed for gearing that I can recall is that described 
by Mr. Geyelin in the Club " Proceedings " of June, 1894. The mortise 
bevels had a peripheral velocity of 3900 feet per minute, but the pressure 
per inch of face was only about 680 pounds, the diameter and speed being 
made high to reduce the pressure on the teeth. I understand that the life- 
time of these bevels is not long. If made of a grade of steel, as previously 
described, their diameter and speed could be considerably reduced and pro- 
longed endurance would be realized. 

About the same time No. 63 was installed a similar application was made to 
another mill, the gear having a different speed ratio, and the angular velocity 
being lower. See Fig. 64. 

Pinion, Wheel, 

percent. per cent. 

Carbon o . 90 o . 60 

Manganese o . 64 o . 64 

A much larger set had been previously employed, transmitting, about 2400 
horse-power at 750 feet per minute peripheral speed, involving a pressure 
per inch of face of 3500 pounds. This latter pair were 4 feet and 8 feet re- 
spectively, 7^2 -inch pitch, 30-inch face, cut with involute teeth of 14 degrees 
obliquity. See Fig. 65. 

Pinion Gear 

Carbon °-52 0.42 

Manganese 0.55 o . 73 

Silicon o. 107 o. 279 

Phosphorus 0.022 0.078 

Sulphur 0.02 o . 05 

These gears have all rendered excellent service, and to-day are apparently 
as good as at the beginning. 

As considerable expense is involved in cutting large gears of hard steel, it 
is sometimes practicable to rough-cut the gear after it is made as soft as pos- 
sible by slow cooling, a higher degree of hardening being imparted before final 
finishing by air hardening or rapid cooling from the refining heat. This is 
not infrequently done in the case of screws and gears of moderate dimensions. 
In this event it is desirable to have the ratio of manganese low — say, not 
over 0.5 or 0.6 per cent. — as a high manganese content seems to impart a 
permanent hardness that is not reduced by slow cooling. 

It appears to be practicable to maintain sliding surfaces of steel if one of the 



SPEEDS AND POWERS g5 

surfaces is hard, even if the other is comparatively soft, but for steel gearing 
for ordinary purposes I would suggest the use of steel not less than 0.4 carbon. 
If the speeds and pressures are unusually high, a much harder grade of steel 
becomes necessary. When a small pinion engages with a large wheel, the 
former alone can be made of high grade steel approaching to a carbon con- 
tent of 1 per cent. When extreme speeds and pressures become necessary, 
the best results will be found by using in both wheels steel having a carbon 
content approaching 1 per cent., or an equal hardness, obtained by lower 
carbon and high manganese or other desirable hardening addition. With 
gearing accurately cut from steel of this character and securely mounted, it 
is believed that reasonable endurance will be obtained when the product of 
speed and pressure, divided by pitch, each within certain limits, does not 
exceed 1,000,000: for example, a speed of 3,000 feet per minute and 1,600 
pounds per inch of face, or vice versa for gear of 5-inch pitch, assuming, so 
far as we know, a maximum speed of 5000 feet per minute for gear of any 
pitch, and permissible pressure to be proportional to the pitch. 

This statement that speeds and pressures are reciprocal, or as one is in- 
creased the other must be reduced, in a fixed ratio, may not strictly be a 
rational one, but in a broad and general sense it is correct within the usual 
limits of practice. 

It will be understood that such a generalization as herein stated would 
apply to pinions having a liberal and not the minimum number of teeth. 

In the discussion of Mr. Christie's paper, Mr. E. Graves gave particulars of 
three duplicate sets of cast-steel bevel wheels. The pinions are the drivers 
and are 57.39-inch pitch diameter and have 36 teeth, 5-inch pitch, 20-inch 
face. The wheels are 74.8 inches diameter and have 47 teeth. The teeth 
are carefully cut to involute lay-out and are 3.43 inches high. The normal 
speed of the pinion is 360 revolutions per minute, giving 200 revolutions per 
minute to the wheel and nearly 4000 feet circumferential speed on the pitch 
line. The horse-power transmitted is 1300. Assuming the entire load to 
be distributed along the outer end of one tooth, the fiber strain would be 
2,100 pounds per square inch at the root of the tooth. 

The pinion is mounted on the upper end of a 10-inch shaft, 148 feet long, 
with a turbine wheel at the lower end. Both shafts extend through the gears 
and are supported in a massive bridge casting with adjustable bearings. The 
gears are enclosed in a casing and are lubricated with oil fed under pressure 
through several jets applied just in front of the teeth as they mesh together. 

The gears have been in service for five years, but have not been entirely 
satisfactory. Their wearing power in the sense of resisting abrasion is satis- 
factory, but the teeth break. This breakage is confined to the pinion, the 



g6 AMERICAN MACHINIST GEAR BOOK 

nature of the break being the same in all cases, beginning at the large end. 
cracking around the root and following along the tooth. The quality of the 
steel in castings is the ordinary commercial article. The widest variation 
in analysis observed is, in one instance: 

Silicon 0.25 

Sulphur o . 036 

Phosphorus o . 071 

Manganese o . 74 

Carbon 0.31 

Another: 

Silicon 0.27 

Sulphur o . 03 

Phosphorus o. 032 

Manganese o . 80 

Carbon o . 23 

As is to be expected, the softer metal has resisted breaking the longer. In 
two sets of these gears the resisting work is of a varying nature with sudden 
and wide fluctuations; in the third instance the working is more constant. 
This variation of conditions does not seem to have influenced failure, as the 
teeth have broken in all the sets. 

One of the practical difficulties in operating bevel gears of the nature de- 
scribed is the difficulty of holding them so that they will be in proper contact ; 
longitudinal motion in either shaft throws them out of pitch. The most 
serious problem, however, is in securing and maintaining shafts so that the 
extended axis fines of same pass through a common point. The effect of 
power transmission from pinion to gear is to put these axis fines out of posi- 
tion, moving them in opposite directions and resulting in end contact of teeth 
and concentrated load instead of evenly distributing the load along the whole 
length of tooth. In this particular the question of maintaining bevel gears 
is decidedly more of a problem than that of spur gears. In this latter case 
small end motions of carrying shafts produce no effect, while the wearing of 
bearings is only the shifting of pitch fine, and, as it occurs slowly, it will, 
within reasonable limits, adjust itself. 

As a matter of further interest, I will mention that in this same room with 
these gears are three other sets of bevel gear having cut-steel pinions and 
mortise wheel with cast-iron rims. The diameters and ratios of these — 
speeds, mountings, and service — are practically the same as those described 
but the transmission of power is 1,100 instead of 1,300 horse-power. The 



SPEEDS AND POWERS 97 

pinions have 33 teeth, 53^ -inch pitch, with 20-inch face, the teeth being 
planed down to 2 X A -inch thickness on pitch-line. The wheels have 43 teeth. 
These gears have been in service some seven years. None of the pinions 
has ever given way; the wooden teeth in the wheels, however, last only from 
six weeks to two months, an extra rim being kept on hand for refilling and 
replacing. 

Mr. Lewis, in continuing the discussion, said that in regard to the pressures 
carried by gear teeth, Mr. Christie seems to lay down a rule making the 
product of speed and pressure constant. This would reduce the load in 
proportion to the speed, and it seems to me an open question whether that should 
be adhered to or not. I do not think it has been demonstrated how the 
pressure of the teeth should vary with the speed. Some experiments, I think, 
should be made which would indicate that more clearly than has heretofore 
been done. It is interesting to note his remarks regarding the influence of 
the hardness of the metal upon the pressures carried, and instead of reckoning 
the pressure by the inch as so much per inch of face, it seems to me the pitch 
should also be included, because the face of a gear tooth is very much like a 
roller, and the pressure carried by a roller varies with its diameter as well 
as with the face. Some authorities seem to think that it should vary with 
the square root of the diameter, others directly with the diameter, and I am 
inclined to the latter opinion. If gear teeth are proportioned for strength, 
they are also proportioned for wearing pressure and surface to carry the load. 

Mr. W. Trinks said: I wish to call attention to an article on high speed gear- 
ing in the November and December numbers of the "Zeitschrift des Vereins 
deutscher Ingenieure," 1899, by the chief engineer of the General Electric 
Company, at Berlin, Germany. The experiments show that there is no rule 
for the relation between pressure and speed, it depends upon accuracy; the 
load on the teeth may be the higher the more accurately the gears are made. 
A remarkable method of manufacturing gears was the outcome of the ex- 
periment. The curves are laid out on paper three or four times the size of 
the real tooth, reduced to proper size by photography, transferred on sheet 
steel, and etched in. Thus the highest degree of accuracy is obtained. 

It was found that neither cycloidal nor involute curves gave the best results. 
Another curve was developed with a view to reducing the sliding motion 
between the teeth. The article contains very interesting diagrams on this 
point. By dividing the length of two working teeth into an equal number 
of parts, the amount of sliding action can be determined and the fact shown 
that it is reduced to a minimum by these methods. 

Another thing shown by the paper is never to place a flywheel close to a 
gear. If possible, have a good length of shaft between. Slight inaccuracies 



98 AMERICAN MACHINIST GEAR BOOK 

in the pitch of the wheel require acceleration or retardation of the mass, 
and in order to do this force is needed. This force causes a hammering on 
the teeth which may break them — in other words, plenty of elastic material 
should be between the inertia masses and the gears. I feel pretty sure that 
all engineers will be much interested in the article; it is a valuable treatise on 
highspeed gearing. 

Mr. Christie added: The bevel gears described by Mr. Graves are very 
interesting and useful as a record. It is much more difficult to obtain satis- 
factory results with bevels than with plain spurs, as any deviation from correct 
alinement is fatal to correct tooth action in the former. In this instance, 
while speed is very high, the pressure on the teeth is comparatively low — 
about 750 pounds mean pressure per inch of face. Thus the products of speed 
and pressure in relation to the pitch are considerably below the quantity as- 
sumed as a safe maximum. 

Regarding the quality of the material, the manganese is too high. While 
steel of this composition would be moderately hard and wear fairly well, it 
would be somewhat brittle. It is not surprising to learn that some teeth gave 
w r ay by fracture. If the relative proportions of carbon and manganese in the 
steel were reversed, it would be a much better material for the purpose. 

A SPUR GEAR ANGLEMETER * 

In the design of spur gears it is desired to give to the teeth such a form that 
as the gears mesh with each other, the relative motion of the two will be the 
same as that of two cylinders whose diameters have the same ratio one to the 
other as have the diameters of the two pitch circles. By the aid of kinematics 
gear teeth can be so designed as to give exactly this relative motion between 
the gears. However, in the manufacture of gears, factors enter which make 
the form of the teeth of the gears as they come from the shop somewhat differ- 
ent from that developed by kinematics. This variation is quite marked in 
the case of rough-cast gears. Here several errors enter. Because of the 
difficulty and time required in developing a tooth outline according to kine- 
matics, arcs of circles which approximate the correct outline are used in laying 
out the gears in the drafting room. From these slightly inaccurate drawings 
the patternmaker makes the patterns, which are apt to vary slightly in form 
and the spacing of the teeth from the drawing furnished by the draftsman. 
These inaccurate patterns then go to the foundry, and from them the molds 
are made. In making the mold, in order to draw the pattern it is rapped loose, 
so that the mold is slightly larger than the pattern and still more inaccurate 

*W. M. Wilson, American Machinist, April 13, 1905. 



SPEEDS AND POWERS 99 

in outline. The casting is poured, and in cooling is warped out of shape 
because of the cooling stresses leaving a gear with a final error in the form of 
its teeth made up of several smaller errors, as just enumerated. 

A realization of the presence of these errors suggested that a knowledge of the 
final inaccuracy in the forms of the teeth of rough-cast gears would be of interest 
and perhaps not without value. With this idea in mind, an anglemeter for 
determining the variation in the angular velocity ratio of gears was designed 
and built. 

The anglemeter consists mainly of a responsive frame carrying a drum on 
which is wrapped a card. If these gears operate without vibration the pointer 
would draw a straight line around the card. Figs. 66 and 67 will be practically 
self-explanatory. The pointer was found to multiply the actual variations 
9.4 times. 

TEST OF A PAIR OF CAST GEARS 

The instrument was connected to a pair of spur gears, No. 12020, each 
having a circular pitch of 1 Y A inches and 20 teeth. Each gear was mounted 
on a 2y|-mch shaft. As the two gears and the two shafts were the same 
size, no reducing motion was needed. The guides of the instrument were 
not long enough to allow a complete revolution of the gears, so the teeth were 
numbered from 1 to 20 and one card drawn for teeth 1 to n and another for 
teeth from n to 1 . The points on the curves corresponding to the time when 
these teeth came into contact were marked and then the cards were cut to 
these marks and pasted end to end as shown in Fig. 66. 

The details of the design of this pair of gears were not known, but judging 
from the shape of the teeth they were modifications of involutes and the out- 
lines of the teeth were probably laid out according to some empirical method 
in general use in the manufacture of rough-cast gears. Before the test the 
gears had been run long enough for the bearing surface to become smooth. 

Cards were taken when the gears were adjusted at different distances apart. 
If the gears had been true involutes the velocity ratio should be constant 
and the same for the different distances between the centers. Whether the 
velocity ratio were constant or not is readily seen from the curves in Fig. 66. 
Due to the uncertainty of the shrinkage of castings, the diameter of the gears 
was not exactly equal to the computed diameter, but a trifle greater. Judging 
from the clearance at the ends of the teeth the proper distance between the 
centers of the gears was 9^8 inches. Card A , Fig. 66, gives a curve correspond- 
ing to this distance between centers. Two curves were drawn before the 
card was removed and the ability of the instrument to duplicate a curve is 
considered as evidence of its accuracy. It is seen from the curve that the 



IOO 



AMERICAN MACHINIST GEAR BOOK 



velocity ratio instead of being constant is quite erratic in its variations. While 
in general the variations in the curve are not related to each other in any way, 
yet for a portion of the card at least the waves in the curve correspond roughly 
with the teeth on the gears. This is especially true of the portion of the curve 
corresponding to the teeth from n to 20. The curves on Card B were drawn 
when the distance between centers was 9-J-f- inches. In this case there is 
a more marked relation between the waves in the curves and the teeth on the 





10 /16 between Centers 
FIG. 66. CURVES FROM CAST GEARS. 



gear than in the former case. For Card C the distance between centers was 
io T V inches, and the regularity of the waves is still more marked. In this 
case the card was taken for half a revolution only. 

For all the cards the arrows above the different curves indicate the general 
direction of the tangent to the curves which give the angular acceleration of 
the driven as compared with the driver. 

From the curves shown the following conclusions have been drawn relative 
to the forms of the teeth: 

(1) The fact that the relation between waves in the curves and the teeth 
of the gears is increased indicates that all the teeth are subject to a common 
error due to the empirical method in which the outlines of the teeth were 
developed, also that this error causes a greater variation in the velocity ratio 
when the distance between the centers of the gears is increased. (This is in 



SPEEDS AND POWERS IOI 

accordance with the empirical method presented in Kent's hand book and at- 
tributed to Molsworth. By using the method referred to, the resulting tooth 
outline would be wider below and narrower above the pitch circle than true 
involute teeth. When the distance between centers is normal these errors 
annul each other almost completely, but as the distance is increased the result 
or the errors is more apparent.) 

(2) Irregularities in the curves indicate errors peculiar to the individual 
teeth, which evidently are involved in the progress of patternmaking and 
molding. 

(3) A shifting of the general vertical position of the curves on the cards, as 
shown at teeth 4 and 14, indicates errors in the spacing of the teeth. 

An effort has been made to analyze the curve in Card A. The portion of 
this curve ABC has been chosen as being a wave whose relation to an indi- 
vidual tooth is the most evident. The abscissa A C represents the angular 
space occupied by one tooth on the gear, and A B and B C each represent one- 
half of that angular space. The ordinate between A and B measures 0.18 
inch, and that between B and C measures o.n inch. Dividing by 9.4, the 
constant for the instrument, it is found that the driven gear gains on the 
driver by an angle whose arc measured on the circumference of the shaft is 
0.18 divided by 9.4 equals 0.019 inch, while the gear turns through an angle 

equal to X l A = 9 degrees. 

20 



An angle whose tangent is 

.019 



is 45 minutes. 



1.46885 (radius of shaft) 

Computing in the same manner the angle corresponding to the ordinate 
B C is found to be 30 minutes. That is, while the tooth No. 16 is in action 
the driven gains relative to the driver by an angle of 45 minutes while the latter 
is turning through an angle of 9 degrees, and then loses by an amount of 30 
minutes in the same space. An error of 0.019 inch measured on the circum- 
ference of the shaft corresponds to 0.06 inch measured on the pitch circle. 

TEST OF SPECIAL GEARS 

The second pair of gears to which the instrument was attached consisted of 
a No. 1 2016 pinion having ij4 inches circular pitch and 16 teeth and a No. 
12050 gear having 50 teeth. The gears were designed to mesh properly 
when the distance between centers varies by an amount equal to }i inch. 
The shrinkage of the casting was not as much as was expected, so that the 
normal distance between centers is 0.20 inch more than the sum of the com- 



102 AMERICAN MACHINIST GEAR BOOK 

puted radii. If the gears were true involutes the velocity ratio should be 
constant when the distance between centers of the gears varies from 15.7 to 
16.2 inches. 

The outlines of the teeth of the gears are involutes having an angle of 
obliquity of 23^ degrees when the distance between centers of the gears is 
normal. The mold for No. 12016 was made from a pattern, while for No. 
12050 it was made from a tooth-block as used in a Walker gear-molding ma- 
chine. The pattern and tooth-block were both made from drawings laid out 
in the drafting room. The tooth outlines for these drawings were developed 
according to kinematics, so that for the exception of any small error which 
the patternmaker might make, the teeth on the pattern and tooth-blocks are 
correct involutes. 

How near the castings approached to involute gears is shown by the curves 
in Fig. 67. As the two gears were not of the same size, it was necessary in 
taking the cards to use some means of making the average velocity of the two 
carriages of the instrument the same. To do this the hub of the gear was 
turned to a smooth surface and the end of the shaft of the pinion was turned 
down until its diameter bore the same ratio to the diameter of the hub of the 
gear that the diameter of the pitch circle of the pinion bore to the diameter 
of the pitch circle of the gear. This gave a rigid and accurate reducing motion. 
The method of taking the cards was the same as for the other pair of gears 
except that a curve for a complete revolution of the pinion was obtained upon 
a single card. The distance between centers of the gears for the different 
cards is as indicated in Fig. 67. Cards D, E, and F cover the range of varia- 
tion in the distance between centers for which the gears were designed. From 
these curves it is seen that there are no waves on the curves corresponding 
to the individual teeth, the curves for the most part are smooth, and are 
practically the same for the different distances between centers of the gears. 
For Card G the distance between centers was made 16.45 inches, or .75 inch 
more than the least distance between centers at which the gears were sup- 
posed to run. The remarks made in regard to curves D, E, and F apply 
equally well to this curve also. For cards H and / the distance between 
centers was so great that the point of contact of the teeth did not follow the 
line of obliquity throughout the angle of contact but lay outside of this line 
for the latter portion of the period of action of the teeth. 

This is clearly brought out in the forms of the curves in which waves cor- 
responding to individual teeth are quite evident. 

It will be noticed that for all of the curves for the second pair of gears there 
is an extended wave near the middle of the card and also near each end, so 
that if several cards were placed end to end so as to form a continuous curve 



SPEEDS AND POWERS 



103 




15.7 " between Centers 




16.2 " between Centers 




16.45 " between Centers 




16 15 14 13 12 11 10\^ 9 



7 6 5 4 3 2 1 





16.75 "between Centers 




16.82" between Centers 
FIG. 67. CURVES FROM SPECIAL GEARS. 



104 AMERICAN MACHINIST GEAR BOOK 

there would be two complete waves on this curve per revolution of the pinion. 
This indicates that the pinion instead of being exactly round is oval in form. 
This deformation can be traced to the molder in the foundry who, in order to 
draw the pattern, raps it loose from the sand and unless great care is taken 
increases the diameter more in one direction than in the other. The wave in 
the curve is traced to the pinion instead of to the gear because the spacing of 
the teeth for the gear was done by means of a molding machine and any errors 
which might occur would be peculiar to the individual teeth instead of being 
common to a group of teeth and gradually increasing and decreasing from 
zero up to a maximum and then back again to zero, as indicated on the 
cards. 

From the curves in Fig. 67 the following conclusions are drawn relative 
to this pair of gears: 

1. The outlines of the individual teeth are true involutes. 

2. The spacing of the individual teeth on both the gear and the pinion is 
accurate. 

3. The pinion is slightly oval in form. 

4. For the gears used the angular velocity ratio is constant through a range 
in the variations in the distance between centers equal to 0.75 inches. 

GENERAL CONCLUSIONS 

If one would be allowed to draw conclusions from the tests of two pairs of 
gears only, the following statements might be made in regard to the errors 
in the teeth of rough -cast gears: 

1. Of all the errors the greatest is due to laying out the teeth in the drafting 
room. 

2. Given an accurate drawing the patternmaker can make a pattern very 
accurate in form. 

3. In using the patterns the molder is apt to make the mold slightly oval 
in form, but in using a molding machine the error induced in the foundry is 
very small. 

4. The surfaces of the teeth of cast gears are so smooth as not to affect the 
angular velocity ratio. 

The difference in the appearance of the curves for the two pairs of gears is 
very much in accordance with the manner in which they meshed, the second 
pair running with very much less noise than the first pair. At a speed of 250 
revolutions per minute the first pair almost threatened to shake the testing 
machine to pieces, while at the same speed the second pair ran with but little 
vibration. 



SPEEDS AND POWERS 



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AMERICAN MACHINIST GEAR BOOK 



It might be said that the patterns for the second pair of gears were com- 
pleted before there had been any thought of subjecting the gears to a test, 
which fact eliminates the possibility of unusual care having been taken in mak- 
ing the patterns. 

GEAR EFFICIENCY 

No thorough investigation has yet been made of the efficiency of gears, and 
very few data of any sort has been published, therefore little can be said. One 
thing is tolerably well known: the efficiency of a gear drive varies with its 
ratio — that is, a reduction of 2 to i will be more efficient than one of 10 to 1, 
other features governing the efficiency being the accuracy with which the 
teeth are formed, and spaced, the arc and obliquity of action, and the con- 
dition of the engaging surfaces. It is generally conceded that the length of 
face does not affect the efficiency. 



22 X A degree 


involutes. 


2434 degree involutes. 


i9 T /< degree involutes. 


16" between 


centers. 


16.2"" between centers. 


15.7" betw 


een centers. 


Tangential 


force at pitch 


Tangential force at pitch 


Tangential force at pitch 


line, 36 


1 lbs. 


line, 


355 lbs. 


line, 


353 lbs. 


R. P. M. 


Efficiency, % 


R. P. M. 


Efficiency, % 


R. P. M. 


Efficiency, % 


88 


92.40 


116 


91.00 


no 


00.00 


123 


90.00 


68 


96.90 


69 


94,40 


140 


89.90 


90 


91.50 


127 


86.80 


156 


91.30 


140 


89.90 


147 


88.90 


172 


91,30 


163 


90.10 


162 


88.80 


188 


92.30 


177 


90.00 


196 


88.90 


216 


92.30 


197 


99.90 


218 


89.20 


229 


92.40 


208 


90.00 




Av. 89.60 




Av. 91.50 


214 


90.30 
Av. 91.20 






Efficiency of Gears 


Efficiency of Gears 


Efficiency of Gears 


alone, 


92.30% 


alone 


, 92.10% 


alone 


, 90-50% 



Table 18 — Gear No. 12016 Driving 12050 — Lubricated 



This loss comes mainly through the sliding action of the teeth when entering 
and leaving contact; it would seem, therefore, that the arc of action should be 
as short as possible, carrying the load of that portion of the tooth where it 
can be best borne, also, as there is less friction in the arc of recess than in the 
arc of approach, it will increase the efficiency to lengthen the addendum of 
the driving gear. The contact, however, should be for an equal distance above 
and below the pitch line to avoid extreme sliding friction. If the entire ac- 
tion was rolling contact, there would be little loss. 

Tables 17 and 18 were originally published in American Machinist, by W. 
M. Wilson. They give the result of experiments with the same case tooth 



SPEEDS AND POWERS 1 07 

spur gears mentioned on pages 99 and 101. The gears in Table 18 were 
made with extra long teeth, the various angles of obliquity as given were 
obtained by adjusting the gear centers. 

The result of these tests is summed up as follows: 

1. The efficiency of rough gray iron spur gears is independent of the speed 
of the gears within the range covered by this report. 

2. There is no indication from the limited number of tests that the amount 
of power transmitted affects the efficiency to any great extent. 

3. The use of a heavy grease on the teeth increases the efficiency slightly 
(average from tests 1.7 per cent.). 

EFFICIENCY OF LARGE GEARS 

Many of us know things that are not so, and with some of us a part of this 
useless knowledge may be in regard to the efficiency of large gears. 

We believe that we are well within the bounds of truth in saying that the 
majority believe that large gears are very inefficient. 

This misinformation or lack of information is easily explained. The data 
that are available in regard to the efficiency of gears of any size are few at best 
and apply to pinions and small gears. Large gears have not been extensively 
tested, for there are but few technical schools and factories equipped with 
appliances for gear testing and capable of absorbing several thousand or even 
several hundred horse-power. Again we do not always distinguish between 
a large gear with cast teeth and a similar gear with cut teeth. 

In the early days of factory engineering, gear drives were common for trans- 
mitting power from shaft to shaft throughout the plant. As time went on 
these drives were replaced — in some cases by belting, in other cases by a change 
from mechanical transmission to electrical distribution of power. It has 
been very easy to assume that the reason for discarding the gears was because 
of their inefficiency. Enthusiastic advocates of electric drive have time and 
again referred to the substitution of motors for mill gearing with elation and 
have either stated or implied that the change brought about a great saving of 
power. 

In further support of the common belief that gearing in large sizes is inef- 
ficient we quote the following from the presidential address of Mr. Denny to 
the Institution of Marine Engineers, in England, on October 5, 1908: "It 
has frequently been suggested that if some inspired engineer could evolve a 
system of gearing that would be lasting and reliable, not too noisy, and would 
not absorb in friction more than, say, 10 per cent, of the power, turbine engines 
would be capable of application to any speed of vessel and to any size of 
propeller." Here Mr. Denny gives expression to an oft-repeated suggestion, 



io8 AMERICAN MACHINIST GEAR BOOK 

that if large gearing could be made having an efficiency of 90 per cent., a big 
step forward would be made. 

With direct bearing on this belief we have the record of performance of the 
largest gears in point of horse-power ever made and tested — the Melville-Macal- 
pine reduction gear, through which 6000 horse-power has been transmitted. 

To show the measure of the performance we quote a paragraph from an 
article by George Westinghouse in the January number of The Electric Journal: 
" Considering the important bearing of the question of efficiency on the ulti- 
mate success or failure of the gear, it is peculiarly gratifying to have found by 
repeated trial and careful measurement that the transmission loss hoped for 
by Mr. Denny has been divided by seven. To be exact, the efficiency sur- 
passes the more than satisfactory figure of 98.5 per cent." 

The hope was for an efficiency of 90 per cent., the fulfilment was an efficiency 
of 98.5 per cent. 

Large bevel-gear drives have perhaps been especially condemned; it is, 
therefore, of interest to read the following from a paper presented by Prof. 
CM. Allen to the American Society of Mechanical Engineers on the testing 
of water wheels: "The total horse-power delivered to the generator was ap- 
proximately 700. The driving gear was of the ordinary wood-mortise type, 
outside diameter 6 feet 5 inches approximately, with 68 teeth 14 inches wide, 
meshing with a cast-iron pinion which had 48 teeth with planed-tooth outline. 
At full load the loss of the gear was 3.5 per cent, and 3.4 per cent, for two 
separate units, or the efficiency of the horizontal-shaft vertical-wheel gear 
drive was about 96.5 per cent. The gears were well lubricated with a thick 
grease. 

"About nine months later it was necessary to test one of these same units 
in exactly the same manner. The loss in gears this time was a trifle less, the 
test giving 3.1 per cent. 

"All of the information obtained concerning the loss due to bevel-gear 
drives leads the writer to conclude that if gears are properly designed, set up, 
and operated, and are not overloaded intermittently or continuously or left 
to care for themselves, they should show an efficiency of from 95 to 97 per cent." 

Are not these bevel-gear drives efficient compared with the majority of 
mechanical devices? 

As a matter of fact, do not the above figures agree with common sense? 
The quantity of heat generated by the absorption of one horse-power for an 
hour is 2545 British thermal units. If we try to give an expression to the 
statement that large gears are inefficient by assuming a loss of 10 per cent., 
in the case cited by Professor Allen, 70 horse-power would be dissipated in 
heat. A multiplication will show what this means in British ^ thermal units 



SPEEDS AND POWERS 



109 



per hour, and the presumption is strong that the gears would heat, cut, and 
wear under such a condition. 

The same reasoning will probably apply to many large-gear drives con- 
cerning which there has been speculation in regard to efficiency. Had they 
been extremely inefficient, they would not have operated satisfactorily for a 
long period of time. 

These figures show that large-gear drives are not necessarily inefficient, 
but, on the contrary, may be decidedly the reverse. 



SECTION IV 

Gear Proportions and Details of Design 

These formulas should not be used indiscriminately, as one design will not 
meet all conditions. No attempt has been made to proportion arms or rim 
to the power to be transmitted, as all proportions are derived from the pitch 
and face of the gear, with the double object of obtaining equal strength and 
sound castings. 

As the smallest section of a casting is, per square inch, its strongest part, 
this fact should enter more into the question of proportion than has been the 
custom in the matter of gears. To illustrate the value of this, a test piece cut 
from the point of a cast gear tooth will often be as much as 30 per cent, stronger 
than a similar piece cut from its root. Hence the rim of a gear is made thinner 
than has been the practice, and the central rib deeper, to secure the necessary 



Taper V 2 Inch 

per Foot, t— 



W' 




FIG. 68. PLAN AND SECTION OF A SPUE. GEAR, SHOWING NOTATION. 

section and thus obtain a stronger casting with the same weight of material. 
This same rule applies to the hub, the outside diameter being reduced and a 
deeper rim added, and reinforcements placed over keyways, or, better still, 
two reinforcements directly opposite, especially if the gear is to be balanced; 
in fact, this is imperative even at relatively low velocities. For the same 

no 



GEAR PROPORTIONS ill 

reason the teeth should be cored whenever it is possible to do so, as the rim 
of a casting for a cut gear has always the heaviest section, and, therefore, 
most subject to blow holes, especially at the junction of the arms and rim. 
It follows, then, if this be true, that the more uniform the section throughout, 
the sounder and stronger the casting. 

Cored teeth, however, should be machine spaced, as uneven spacing will 
render it difficult, if not impossible, to cut the teeth in the usual manner. To 
secure accurate spacing when cutting, it is absolutely necessary that the cutter 
have an equal amount of stock to remove from each side of the tooth space. 
Also exposure to hard scale and core sand quickly destroy the cutter. When 
the teeth are to be planed this point is not of so much importance, as a little 
unevenness of stock can be readily taken care of, but the cut should be always 
under the scale if time is of any importance. 

FORMULAS 

These formulas are based on Brown & Sharpe standard 14K -degree in- 
volute tooth. 

Thickness of Rim, M = 3 ' 927 , or 1.25 p' 

P 

Mean Thickness of Rim, M' = — , or 1.60 p 1 

P F 



2.vj 

Mean Thickness of Rim under Tooth, R' = — ^ , or 0.913 p' 

Whole Depth of Tooth, W = 2 ' 157 , or 0.6866 p' 

P 

Minimum Thickness under Tooth, R' = — — — , or 0.563 p' 

Area of Rim Total, MF' = M'F 

Area of Rim under Tooth = R'F 

Average Area of Arm, JE = t F 1.3, or MF 0.52 

Average Thickness of Arm, A = \ / 1 ' 2 ' 

V 3 

Average Width of Arm, E = 3 A 



Outside Diameter of Hub = Bore + % \ f NF 

Number of Arms = 4, 6, 8, 10, etc., according to design and diameter of gear. 

p = Diametral pitch. 

p'= Circular pitch. 

/ = Thickness of tooth at pitch line* 



112 AMERICAN MACHINIST GEAR BOOK 



DISCUSSION OF FORMULAS 



Thickness of Rim, M — The thickness of rim should be equal to 3.927 
divided by the diametral pitch, or 1.25 multiplied by the circular pitch. When 
the gear is small and accurately made it is often good practice to make 
the dimension 1.12 of the circular pitch, and so secure the same section 
and additional strength by adding 50 per cent, to the depth of the central 
rib. 

Mean Thickness of Rim, M' — By mean thickness is meant the thickness of 
one side of a parallelogram necessary to contain the actual area of the rim. 
This will take care of the central rib and fillets, so that by multiplying the 
width of the face of the gear by dimension M' the entire area of rim may be 
obtained. This will be found necessary also for estimating the weight. 

Mean Thickness of Rim under Tooth, R! — This dimension I — : J multi- 
plied by the width of the face will give the area of the entire section of the rim 
and rib under the teeth. 

Minimum Thickness of Rim under Teeth, R — This determines the thickness 
of the rim under the tooth measured at the edge of the rim. 

Whole Depth of Tooth, W — This gives the whole depth of the tooth as per 

2.1 ^7 
Brown & Sharpe standard = 0.6866 p', or — . 

P 

Total Area of Rim — The total area of the rim is found by multiplying the 
mean thickness M' by face of gear F. 

Area of Rim under Tooth — The area of the rim under the tooth is deter- 
mined by multiplying the mean thickness R r by the face F. 

Average Area of Arm, JE — The average area of the arm is that area midway 
between the inside of the rim and the outside of the hub, and is found by add- 
ing 30 per cent, to the area of the tooth at the pitch line, or the thickness of 
tooth at pitch line t X F X 1.3. The same result may be reached by taking 
0.52 of M ' X F, although the foregoing is simpler. Taper of arm to be H 
inch per foot above and below this point. 

Average Thickness of Arm — The average thickness of arm (A) may be de- 
termined by dividing the quotient of the area of arm MY 1.27 by 3, and 
extracting the square root. If the arm was made in the form of a parallelo- 
gram it would not be necessary to multiply the area by 1.27, but as it is to 
be elliptical, this is essential to insure sufficient section, as 27 per cent, of the 
area of the parallelogram is lost when inscribing an ellipse therin. 

Note. — If width of arm is desired, 2 or 2^ times the thickness instead of 3 times, as 
given, 2 or 23^ is to be substituted. 



GEAR PROPORTIONS 



"3 



Average Width of Arm, E — To determine the width of an arm multiply its 
thickness by three. 

Outside Diameter of Hub — As a rule the outside diameter of a gear hub 
is made double that of its bore, but when key ways are reinforced, or the gear 
is to carry a load less than proportional to the diameter of its shaft, the hub 
diameter may be less than this rule prescribes. Thus, if a gear of 30-inch 
pitch diameter was mounted on a shaft 15 inches in diameter, the hub diam- 
eter should only be increased sufficiently to maintain its section and strength 



1.45 



1.40 



1.35 



1.30 



1.25 



1 
3 



1.20 



1.15 



-SJ 1.10 



1.05 



1.00 



0.95 



0.90 






30 40 50 60 70 



90 100 110 120 130 140 150 160 170 180 190 200 210 

Number of Teeth 



CHART 5. MULTIPLIER FOR INCREASED NUMBER OF TEETH OF SPUR GEARS. 



proportional to the gear, not to the shaft. The formula given will propor- 
tion the hub to easily carry the entire load applied to the gear but should be 
used with discretion. Generally, however, when the bore is small or propor- 
tioned to the diameter of the gear, the outside diameter of the hub may be 
taken as 1.75 times the diameter of the bore with reinforcement for key ways. 

From the foregoing it is obviously almost as important that the hub should 
not be disproportionately heavy as that it should be heavy enough. But if 
for any reason a materially heavier hub is required, it should be split, by 
means of thin cores, into as many equal, radial sections as there are arms in 
the gear, thus obviating blow-holes, and strains caused by shrinking. Fill the 
cored spaces with babbitt or lead before machining, and shrink steel bands 
on the hub for a grip on the shaft. See Fig. 69. 

Number of Arms — There is no definite rule for this, as this point depends 



H4 



AMERICAN MACHINIST GEAR BOOK 



almost entirely upon the judgment of the designer. In general, however, 
gears up to 60 inches are either webbed or with four or six arms to suit con- 
ditions; over 60 inches eight arms are generally used and over 80 inches in 




160 Teeth. 
3 Inch Pitch 
20 Inch Face 

FIG. 69. PLAN AND SECTION OF SPUR GEAR WITH SPLIT HUB. 



diameter 10 arms. In no case should the greatest distance between the 
arms exceed the length of the arm measured from the center of the gear to its 
intersection with the rim. 

Width of the Face — The face of spur gear is generally estimated at two or 
three times its circular pitch, as follows: 



1 

1^ 

iX 

2 

3 
4 

5 
6 

8 

10 

12 

14 
16 
18 
20 



diametra 
diametra 
diametra 
diametra 
diametra 
diametra 
diametra 
diametra 
diametra 
diametra 
diametra 
diametra 
diametra 
diametra 
diametra 
diametra 
diametra 



pitch 9 inches face 

pitch 7H3 inches face 

pitch 6 inches face 

pitch 5K inches face 



pitch, 
pitch, 
pitch, 
pitch, 
pitch, 
pitch, 
pitch, 
pitch, 
pitch, 
pitch, 
pitch, 
pitch, 
pitch. 



5 
4 

3 
2 

iM 
1V2 

iM 

1 

M 

Vs 

} 

3/ 
3/ 



inches face 

inches face 

inches face 

inches face 

inches face 

inches face 

i inches face 

inches face 

i inches face 

i inches face 

i inches face 

inches face 

inches face 



GEAR PROPORTIONS 



115 



It is becoming better understood, however, that a wider face is more effi- 
cient (" increasing the face does not increase the friction of the teeth in pro- 
portion"),* and as the wear of the teeth is governed by the diameter of the 
gear, or rather by the combination of diameters and the width of the face, 
the face, therefore, should be amply wide, and the pitch just sufficient to resist 
fracture. 

Street-railway gears are made 3-pitch, 5-inch face, with good results. A 
gear face of five times its circular pitch is now generally considered to be good 
proportion. 

WEBBED SPUR GEARS 

No definite rule can be laid down for the design of webbed gears. See Fig. 
70. Generally the thickness of the web is made equal to R", which is the 
thickness of the rim at its thickest part. 

Core holes H tend to make a sounder casting, and furnish means to secure 
the gear while machining. When these holes are made large, or are shaped to 
follow outlines of the arms and rim, ribs are added on each side. Care should 




FIG. 70. PLAN AND SECTION OF WEBBED SPUR GEAR. 



be exercised, however, not to make the arm too light, for when the hub is 
heavy the light section connecting the rim and hub will cool too rapidly, 
setting up serious shrinkage strains, and causing flaws in the casting that can- 
not be remedied by annealing. Sharp corners, small fillets, and narrow ribs 
should be avoided for the same reason. 

* George B. Grant. 



n6 



AMERICAN MACHINIST GEAR BOOK 



SPLIT SPUE. GEARS 

It is not good practice to split a gear between the arms, but when this is 
necessary, the following points should be kept in mind (see Fig. 71): 

Bolts should be placed as close to the 
rim as possible. 

The dimension b must in no case be 
less than the dimension of a; otherwise 
the bolts will be subject to other than 
the direct tensile stress tending to spread 
the gear. 

Section C-C should be stiff enough to 
resist any strain tending to bend the lugs. 
By placing the bolt close in the corner of 
the rim and the lug, the length of the lug 
may be reduced, as this lug need only 




DIAGRAM ILLUSTRATING SPLITTING OF 
GEARS AT THE Rlil. 



be long enough to counteract leverage on 
bolt. 



The bolts should be sirfhciently heavy to earn' the load applied at the 
pitch line of gear tooth, not neglecting the initial stress set up by the tighten- 
ing of the nut, which is generally neglected, sometimes disastrously. When a 




wmm 






/7mn 



kO 



Teeth. 
Inch Pitch. Y 
5 Inch Face. 



o 



o 



or 



FIG. 73. PLAN AND SECTION OF AVERAGE DESIGN OF SPLIT RAILWAY GEAR. 



bolt is placed close in the corner it is necessary to use a stud bolt, drawing up 
the nuts as the gear halves are brought together. 

When the dimension b is shorter than a, Fig. 71, as is generally the case, the 
load on teeth of gear will cause a fracture of the rim, as illustrated in Fig. 72. 



GEAR PROPORTIONS 



117 



The best method of splitting gears is through the arm, as illustrated in Fig. 
73, which is a cut of the type used on street railways. When the gear is large, 
Fig. 74 illustrates a good average design. 

When splitting a gear of an odd number of teeth, the split should be made 
\i of the circular pitch off the center line. This will bring the split through 
center of two tooth spaces. It is good practice to spline the adjoining sur- 




1 



dlhr 



WMM 



m 



FIG. 74. PLAN AND SECTION OF AVERAGE DESIGN OF LARGE SPLIT GEAR. 

faces, as illustrated in Fig. 73, instead of using fitted bolts or depending on 
dowel pins. A spline M inch wide and H inch high will answer for any but 
the largest gears. 

One point that must be considered in designing split gears for high speed is 
the fact that weight at any point or part of the rim, and not integral to the 
rim proper (as lugs for bolting, see Fig. 71), locates the bending moment, and 
if safe speed is exceeded to the moment of fracture, it will occur at or near 
such weight. 

This fact is pointed and explained by Charles H. Benjamin, in an article 
in the American Machinist of December 26, 1901, entitled "The Bursting 
of Small Cast-iron Flywheels." 



I-SHAPED ARMS 



For gears such as are shown in Fig. 69, the proportion of rim, arms, and hub 
may be determined according to formulas given above, the area of arms, of 



n8 



AMERICAN MACHINIST GEAR BOOK 



course, being contained in an I instead of an elliptical section. For gears of 
this size, however, there is more variation in the design of the gear, and the 

values given cannot be followed so closely. 

The I-section arm is much more desirable 
in heavy gears because it distributes the 
metal contained in a section of the arm over 
a greater surface of the rim at their inter- 
section than does the elliptical arm, and, 
therefore, lessens liability of blow holes at 
this point. Aside from this, it gives better 
support to rim in case the face is wide, and 
makes a stronger section than the elliptical 
arm of the same weight. 
For the above reasons I am disposed to advocate a connecting-rod section, 
such as is illustrated by Fig. 75, for smaller gears. 




FIG. 75. SUGGESTED CONNECT- 
ING-ROD SECTION. 



CONNECTING-ROD-ARM SPUR GEAR 

This design could be applied to gears of all sizes: As the face increased, the 
section could be changed — as per dotted lines. This, however, would allow a 
central rib instead of two ribs on each side of the rim as when the hh is turned 




80 Teeth, 6 Inch Pitch, 14 Inch Face, 15H 
Inch Bore, Cast Steel 27,000 Pounds, Rough 
Weight 23,000 Pounds with Cored Teeth. 

FIG. 76. PLAN AND SECTION OF LARGE SPUR GEAR WITH CONNECTING-ROD ARMS. 



the other way (see Fig. 69), but would allow the use of bolts instead of links 

when a split gear had no hub projections. This design is illustrated by Fig. 76. 

For gears of an extremely wide face it will be found that the formula for arm 



GEAR PROPORTIONS 



119 



will give a section that cannot be contained in the space between hub and rim. 
This practically means making a web gear. However, when the face is wide 
it would seem better to use a light web, say, according to dimension on Fig. 
70 and extending ribs toward the side, making a cross-sectioned arm, or using 
the section shown by Fig. 69. This section is generally the most desirable, 
but this depends upon conditions. 

The formulas given here will form a basis for the design of worm and bevel 
gears, although I believe that the + or cross-shaped section is superior to the 
oval arm for worm gears on account of the side strain encountered. 

FOR CALCULATING WEIGHT 

The accompanying Chart 6 gives a rapid, approximate method of calculating 
the weight of a cast-iron spur gear blank for a cut gear, designed according 
to the above formulas. 

This table was derived from a formula by Reuleaux, which gives the weight 
of a cast-tooth gear from the combined product of a constant for the number 
of teeth, face and square of the circular pitch, as follows: 

" The approximate weight of gear wheels, W, may be obtained from the 
following: 

" W = 0.0357 b c 2 (6.25 N + 0.04 TV 2 ), where b = face, c = circular pitch, 
N = number of teeth, and W = weight of gear. 

" The following table will facilitate the application of the formula; it gives the 

W 

value -t— j for the number of teeth which may be given, and the weight may 

u 

be readily found by multiplying the value in the table by b c 2 : " 



N 





2 


4 


6 


8 


20 


5-04 


5.60 


6.18 


6.77 


7-38 


30 


7-99 


8.61 


9.24 


9.89 


10.52 


40 


11.09 


11.90 


12.59 


13-30 


14.02 


So 


14.74 


15.48 


16.23 


17.00 


17.77 


60 


18.55 


19-35 


20.15 


20.97 


21.80 


70 


22.65 


23-50 


24.36 


25.24 


26.12 


80 


27.02 


2 7-93 


28.85 


29.79 


30.73 


90 


31.69 


32.66 


33-63 


34.62 


35-63 


100 


36-63 


37-67 


38.70 


39-75 


40.81 


120 


47.40 


48.54 


49.69 


50.85 


52.03 


140 


59-3Q 


60.56 


61.82 


63.10 


64.27 


160 


72-35 


73-73 


75-io 


76.39 


77.90 


180 


86.54 


88.03 


89.52 


91.02 


92-54 


200 


101.88 


103.48 


104.98 


106.70 


108.34 


320 


118.36 


120.08 


122.15 


123.52 


125.27 



Example: A gear 50 teeth 2 inches pitch, 4 inches face, we have: be 2 14.74 = 4 X 2 2 X 
14.74 = 235.84, say, 236 pounds. 

Weight of Cast-iron Gearing. Reuleaux. 



120 



AMERICAN MACHINIST GEAR BOOK 



In endeavoring to apply this table to practice it was found that the square 
of the pitch (C 2 ) was not a correct factor, and a separate table was made up 
for it. 

.After repeated trials and corrections in the value of the constant for the 
number of teeth, it was noticed (when close results were finally obtained) that 



3 3 , 

S : : 
3U 

3" 

-. 

m 
O 

12" 

u 
e3 

o ■<■•-» 

5-i 

5 

A 





Curve plotted from Observed Data. 

Curve plotted from Equation. 

p' 2 = 1.58 K 

Weight of Spur Gear Blank (in Pounds ) = 
K x Number of Teeth x Face ( in Inches.) 
The above applies to Cast Iron Blanks. 
For Steel Blanks add 10 Per cent. 



5 6 7 8 

Values of Constant K. 



:: 



11 



12 



13 



CHART 6. RELATIONSHIP BETWEEN CIRCULAR PITCH AND FACTOR K US±-D IN ESTIMATING 

WEIGHTS OF SPUR GEAR BLANKS. 



GEAR PROPORTIONS 12 1 

the values ran parallel with the number of teeth, and were, therefore, dropped, 
changing the value of the constants to the square of the pitch, so called, to 
obtain like results. 

To find the weight, therefore, it is only necessary to find the combined 
product of the number of teeth, face, and a constant given for the pitch: 

Weight = number of teeth X F X K. 

The accompanying Chart 6 gives the value of K. 

This formula has its limitation; it cannot be used for low numbers of teeth, 
varying with the pitch, also for large gears there is a variation in design that 
cannot well be covered by one constant. However, the proper constant may 
be readily determined by trial for different constructions. In general it must 
be used with discretion, but is invaluable as a check. The finished weight is 
found by deducting 30 per cent. 

The following equation seems to give a close approximation to the curve 

of Chart 6 throughout the ordinary working range up to $ Y A inches circular 

pitch: p" 1 = 1.58 K, 

in which p' is the circular pitch and K the constant previously referred to. 

p' 2 
Transposing we have K = JL -^, or substituting in equation for weight of 

1.58 

spur gears, 

p' 2 
Weight = -—=- X number of teeth X face. 
1.58 

Beyond 3^2 inches circular pitch there is considerable variation between the 

curves from the actual data and from the equation. However, as stated, 

the great variations in the design of large gears cannot be cared for by a 

single constant. In using the formula its limitations should be carefully 

understood. As a matter of interest, it might be stated that for 6 inches 

circular pitch the curves are again practically in agreement. 

ARMS FOR SPUR GEARS * 

In deducing a formula for gear arms it is assumed that the thickness of 
the rim is sufficient to distribute the load between the arms; this assumption 
is quite justified, as such a depth is necessary to prevent bending of the rim 
between adjacent arms. By equating the expressions for the tooth strength 
and that of a beam supported at one end and loaded at the other, the general 
expression arrived at is 

p' z R(N — n) 
Z = - — — for circular pitch, and 

So A 

7T 3 R (iV— 7) 

Z = ■ . for diametral pitch, 

Sop 3 A 



* 



Henry Hess. 



122 AMERICAN MACHINIST GEAR BOOK 

when 

Z = modulus resistance of arm cross-section. 
p' = circular pitch. 

p = diametral pitch. 

F 
R = ratio of face width to circular pitch = — T . 

P 
F = face width. 

N = number of gear teeth. 

A = number of arms. 

If it is preferred to use the face width itself, instead of its ratio to the cir- 
cular pitch, then 

p' 2 F (xV-7) 

Z = — — — for circular pitch. 

50,4 

p (A T - 7 ) 

Z = ^—i — for diametral pitch. 

$op 2 A 

By these formulas the dimensions of a gear arm of any section whatever can 
be determined. 

As by far the great majority of cast gear arms are of elliptical cross-section, 
these expressions are reduced by inserting the terms of the modulus of re- 
sistance of an ellipse in which the major axis is double the minor, and the 
formulas become, when E = the thickness of the arm at its base, and 

2 E = the width of the arm at its base; 



E = j/ ( N -7)P' z R . . a/ ( n -7)P' 2 F I for circular 
" 20 A ' 20 A } pitch; 

= 3 / (A 7 - 7) tt3 r _ 3^ / (iV-7)*r 2 F ) for diame- 



20 A p z V 20 A p 2 ) tral pitch. 

To reduce the labor involved by the mathematical solution, Charts 7 and 8 
have been constructed, one for diametral pitches ranging from 10 to 3, and the 
other for circular pitches ranging from 1 to 3 inches; as 3 diametral pitch 
is very nearly equal to i-inch circular pitch, the second chart extends the range 
of the first without a break, so that, between the two, any case likely to arise 
will be taken care of. Diametral pitch is given in Chart 7, as the more general 
practice uses that for small and medium-sized gears, while for large work, 
circular pitch is generally employed, and is therefore used as the basis of 
Chart 8. Two charts are required, as the inclination of the pitch diagonals 
toward the end values would become too slight to admit of accurate reading 
on a single one. 

By tracing the number of teeth from the bottom scale to the pitch diagonal 



GEAR PROPORTIONS 



123 




«MN 



Directions: Trace up from the Number of Teeth to the Pitch Diagonal, then Horizon- 
tally to the Vertical above 300 Teeth, then Parallel with the Nearest Diagonal to the Vertical 
Headed with the Arm Number E; then Trace to the left Horizontally to the Vertical R repre- 
senting the Ratio of Face Width to Circular Pitch R. Take the Nearest Diagonal as the 
Arm Base Thickness E. 

CHART 7. PROPORTIONS OF GEAR ARMS FROM DIAMETRAL PITCH. 



124 



AMERICAN MACHINIST GEAR BOOK 




Directions: Trace up from the Number of Teeth to the Pitch Diagonal, then Hori- 
zontally to the Vertical above 300 Teeth, and Trace Parallel to the Nearest Diagonal to the 
Vertical Headed with Number of Arms; then Trace to the left Horizontally to the Vertical 
R representing the Ratio of Face Width to Circular Pitch. Take Nearest Diagonal as 
Base Thickness of Arm E. 

CHART 8. PROPORTIONS OF GEAR ARMS PROM CIRCULAR PITCH. 



GEAR PROPORTIONS 1 25 

in the main portion of the circular pitch chart and referring the intersection 
to the vertical scale under 8, the value is found of 

(N-7) p* iQrA==s arms< 
20 A 
For any other number of arms this is modified by employing the auxiliary 
portion of the chart at the right, referring the value just found along or be- 
tween the nearest slant lines to intersection with that vertical representing 
the number of arms actually used. By now tracing this hight horizontally 
to the left to that vertical R representing the particular ratio of face width 
to circular pitch employed, and taking a reading from the nearest slant line 
crossing this vertical, the value first found is multiplied by R and the cube 
root extracted. 

Past practice gives a face width between two and three times the circular 
pitch, but as the tendency is now toward a wider face, ratios from i3^ to 4 
are given. 

Concise directions are printed with the charts. Dotted trace lines of the 
following examples are also drawn in: 

Example 1. Given a gear of 100 teeth, 4 diametral pitch, 6 arms and ratio 
of face width to circular pitch = 2 l A. 

Trace on Chart 7 100 teeth up to diagonal for 4 pitch, horizontally to num- 
ber of arms 8, slantwise up to number of arms 6, horizontally to the left to 
ratio 2}4 , which is intersected between % inch and 1 inch; therefore thick- 
ness of arm is to be taken as 1 inch and width as 2 inches. By calculation 
the dimensions are 0.96 inch and 1.98 inch. 

Example 2. Given a gear of 270 teeth, 2-inch circular pitch, 6 arms and 
ratio of face width to circular pitch = 2. 

Trace on Chart 8 as before and find 3M inches full as thickness of gear arm 
at base, and 63^ inches full as width. The calculated dimensions 3.27 inch 
and 6.54 inch agree almost absolutely with the much more quickly obtained 
values by the chart. 

In large arms the designer will frequently prefer a cored section. A sat- 
isfactory one will be that of Fig. 77, in which major and minor axes of both 
core and arm are relatively as 2 to 1. By equating the moduli of resistance 
for solid and hollow elliptical sections of these proportions, it is found that 

D* — d 4 
E 3 = — , in which E is the thickness of the solid arm as obtained by 

chart or formula; d and D are dimensions of the cored arm. See Figs. 77 and 78. 
In order to lessen the work of making the core box by substituting flat sur- 
faces for curved ones, an approximation like Fig. 78 will add but slightly to 
the weight, as is shown by the ellipse dotted in for comparison. 



126 



AMERICAN MACHINIST GEAR BOOK 



The ellipse outlines are formed of circular arcs struck from four centers, 
which will approximate very closely to the true ellipse. The construction 
of the core sides is readily apparent from the sketch. 

The arm taper is stated as i in 32 and 16, respectively, for the arm thick- 
ness and width; this gives a pleasing appearance for a moderately long arm, 




FIG. 77. PROPORTIONS OF HOLLOW ARMS. FIG. 78. 



but it is not a hard-and-fast rule, as a greater or lesser taper may be employed 
to suit the designer's fancy without affecting the strength of the arm, unless 
the taper is made so excessive as to bring the dimensions at the rim down to 
one half of these at the base. 

As the tooth and arm are of the same material, the method is satisfactory 
for all cast gears, but this must not be interpreted to mean that this or any 
other formula will prevent shrinkage strain due to relatively large hubs or very 
heavy rims; where these occur, great care must be exercised in the foundry, 

and it will also not be amiss to add a 
generous amount of metal to the arms. 




flAI 

IS 



One Inch Pitch. 



X I 0.60 

A u 1 a 

a 1 <y 
1 ° 

^0.0625" «j5 

o\ II 

a I S 

"'S 



sl a 



FIG. 79. 



Q A 
PROPORTIONS OF RIM GEARS. 



RIM GEAR PROPORTIONS 

Where steel castings prove inefficient 
for the work intended, forged steel rims, 
designed somewhat as illustrated in Fig. 
79, are used. The center is made of cast 
steel of a heavy pattern, the forged rims 
being shrunk thereon. As this rim ma- 
terial may be obtained of a higher grade 
of steel than is possible in the casting, 
and is free from hidden flaws, also as the 



GEAR PROPORTIONS 



127 



rim is renewable, it is a much better and, in the end, a cheaper proposition. 
The rims are made of a forged billet, which is first pierced and then rolled 
into shape by the same process employed for locomotive tires. 

There are many variations of this design, but it is thought best to make the 
face of the center narrower than the rim on account of possible unevenness in 
fitting, and turn a shoulder on center to bring rim up true, instead of de- 
pending upon parallels or surface plates. Also it will be desirable in many 
cases to replace rim without removing gear from the shaft. This is accom- 
plished by rapidly heating rim by means of a circular gas or oil burner made to 
suit diameter of gear and protected by asbestos cover to localize the heat. 



DESIGN OF BEVEL GEARS 

The rules for the design of spur gears may be applied to helical, herringbone, 
worm, and spiral gears, using the circular pitch as a basis. Also for bevel gears, 
taking the proportions from the large end of the tooth. The average design 
of bevel gears is shown in Fig. 80. The hub should be carried well back of the 

face and connected to the rim by ribs, 
4, 6, 8, or 10 in number, depending 
upon the diameter of the gear. The 
hub should not be carried too far in 
the front, or small end, of the gear, as 
a long hub will make it impossible in 
many cases to cut the teeth, for if 
this hub is carried too far it will in- 
terfere with the operation of the 





FIG. 80. BEVEL GEAR PROPORTIONS 



FIG. 8l. BEVEL GEAR WITH LONG FRONT 
HUB EXTENSION. 



machine. See Fig. 81. A small hub, however, should always be put on the 
front end of the gear, otherwise it will be necessary to counterbore to secure a 
finished bearing. 



128 



AMERICAN MACHINIST GEAR BOOK 



RAWHIDE GEARS 

Rawhide gears are commonly made with brass flanges on either side of the 
face to hold the rawhide in position and to engage the mating gear, unless 
rawhide contact alone is desired. It is practice to speak of the face of the raw- 
hide gear as including these flanges. See Fig. 82. There is no great gain in 

1 1 

\< Engaging-Gear >j 

0.0625 p! I 

for Clearance-^, <~ 



' l;i!iJi'!ii!!i:iiiili!ii!ilii!iiiii 



1 . 1 1 1 1 ; 1 1 1 1 1 1 1 i ! ; > ; 1 J j m ; 1 i i ' 



'Inililiin 



Rivet 



- 



■N 



Keyseat 



SP88S8S 

111 H ll'll i I I 




I 



I I 



h Rawhide ^ \<— 



-F-ace- 



I 

- — n 



FIG. 52. RAWHIDE GE.AR. 



T 



/ 



1 



^.'ll'll'llll'i'll'l'l'll'l'l 1 ■ 

^ill«ilil!iliiili!ilii|iWii 



Rivet 



«liiiiiiiiiiiii!il! 




Keyseat 



- 



- 



WPfSB 

Rawhide ;];■ 

11 ! S 1 





'H! 



_ 



I h 

I ! 



0.3125 p' 1 
— Face ^T"- k- 



r? tength-o ver-all H 

FIG. 83. SHROUDED RAWHIDE GEAR. 



preventing the flanges from coming into contact, and to make a rawhide gear 
without cutting through the flanges, as per Fig. 83, is unnecessary and ex- 
pensive. 

There is no reason why steel flanges cannot be used in place of brass, es- 
pecially for the larger sizes; boiler plate will be found excellent for this. The 
thickness of the flange is something that varies greatly, although %, of the cir- 
cular pitch is a fair average. 

For the larger rawhide gears it is recommended that bolts instead of rivets 
be used, as it is impossible to otherwise draw up a wide-face rawhide gear. 



GEAR PROPORTIONS 



129 



The bolt head may be countersunk so that one side of the gear will be flush. 
It is also sometimes possible to put the nut in a counterbore ; this depends, of 
course, on the design of gear. 

Rawhide gears to run loose on the shaft should be bushed. When it is neces- 
sary to move a rawhide gear on a spline one of the flanges should be made as 



T 



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FIG. 84. BUSHING AND FLANGE IN ONE 
PIECE. 



Pace 



EIG. 85. DESIGN OF RAWHIDE GEAR. 



part of the bush, as per Fig. 84. The usual design of the larger size rawhide 
gear is shown in Fig. 85. 

Rawhide bevel gears are designed similar to Fig. 86. Both ends of the teeth 
must be flanged to facilitate the cutting of the teeth. 



13° 



AMERICAN MACHINIST GEAR BOOK 



of apple 
inserted 



Fiber is often used in place of rawhide, 
but is usually more brittle and has a tend- 
ency to wear the engaging gear, although 
made of a harder material. Fiber gears 
are ordinarily furnished without flanges. 
Gears are also made of laminations of 
rawhide and bronze, or fiber and bronze, 
but not to any great extent. 

A fiber stress of 5000 pounds per 
square inch is amply safe for calculating 
the strength of rawhide gears. 

MORTISE GEARS 

The mortise gear is composed of a cast- 
iron rim, containing cored slots, into 
which wooden teeth are driven. • See Fig. 
87. These gears cannot compare either 
in efficiency or cost to a properly cut cast- 
iron gear, but are still used in many 
places where excessive noise is prohibi- 
tive. The wooden teeth are made either 
or maple, treated in linseed oil and cut to the proper form after being 
in the cored rim; otherwise the spacing of the teeth would be gov- 




RAWHIDE BEVEL GEAR. 




Tooth 



Two^eys, One to 

Drive from each 

Side of Face. 



87. PROPORTIONS OF MORTISE GEAR BASED ON 
I-INCH CIRCULAR PITCH. 



GEAR PROPORTIONS 



131 



erned by the spacing of the cores, which can never be very accurate. Replaced 
teeth must be fitted and shaped by hand. The strength of the mortise gear 
is governed by the thickness of the iron teeth in the engaging gear, which are 
made 0.35 of the pitch instead of 0.5 pitch, as for cut gears. The outside 
diameter of the cored rim is turned to the dedendum diameter that the teeth 
will be set an equal distance from the center. 

KEYSEATS 

The commonly accepted keyseat standard is 
that of Jones and Laughlin, as per Table 19. 
In this the width of the key approximates one- 
quarter the shaft diameter. A square key is 
used, one half being in the gear and one half in 
the shaft. Dimensions for taper keys are in- 
cluded in this table. A taper keyseat, however, 
is nothing more than a means of tightening up a 
poor fit; a gear properly fitted to the shaft will 






KEYSEAT TAPER 








A INCH PER FOOT 


STRAIGHT 


KEYSEAT 


BORE 




SMALLEST 








WIDTH 




WIDTH 


HIGHT 




A 


HIGHT 

"b" 


"A" 


"b" 


IN. IN. 


IN. 


IN. 


IN. 


IN. 


8 to f/ s 


2 


2.L 
32 


2 


I 


M " lA 


i 7 A 


H 


1% 


% 


M " 6% 


1% 


1 9 
3 3 


iM 


7 A 


6M " 6% 


iVs 


17 
32 


iH 


% 


6H " SA 


i l A 


A 


1^ 


H 


5% " sA 


iVs 


2.& 
64 


1% 


% 


sH " 4 7 A 


iH 


2.7 
64 


iM 


% 


4% " 4^6 


*A 


H 


i3^ 


% 


4% " 4%> 


I)i6 


tt 


iY 6 


32 


4A " 3% 


1 


Li 

32 


1 


X A 


3 7 A " 3% 


% 


% 


% 


Li 
3 2 


3 5 A " 3% 


7 A 


1 9 
64 


K 


%, 


3 3 4 " 3% 


% 


±1 
64 


% 


13. 
3 2 


3 l A " 2% 


% 


M 


M 


H 


2% " 2% 


% 


IS. 

64 


% 


11 
32 


2VS " 2% 


A 


Lit 

64 


A 


^6 


2% " 2%, 


% 


X 


% 


a 
32 


2 l A " 1% 


Vi 


1L 

64 


A 


M 


I 7 A " 1% 


% 


9 
6 4 


7 A> 


.I. 

3 2 


i 5 A " * 7 A 


A 


A 


% 


% 


iVs " 1% 


% 


64 


% 


3 2 


i l A " 1 


K 


_3__ 
32 


a 


3^ 



Table 19 — Standard Keyseats 



132 



AMERICAN MACHINIST GEAR BOOK 



not work loose if a straight key is used with clearance on the top. The best 
fit can be spoiled by a little carelessness in driving a taper key, as it is sure to 
make the gear run out if driven the least bit too tight. The taper key is ap- 
plicable to only the heaviest work where 
the mass of metal will prevent any such 
distortion. 

For gears that are to be hardened it 
is important that there be a fillet in 
the corners, the top of the key being 
beveled off to suit, otherwise a crack is 
very liable to start from the sharp cor- 
ner of the keyway. For that matter, 
this would be an excellent plan to adopt 
for all key ways. See Fig. 88. 
For machine tools and automobiles the Woodruff key is generally used. 

~H 

b h 

i i 




FIG. 



FILLETED KEYWAY. 





NO. OF 
KEY 



I 
2 
3 

4 
5 
6 

7 
8 

9 

io 
ii 

12 

A 

*3 
14 
15 



DIAM- 


THICK- 


DEPTH 


ETER 


NESS 


OF 


OF 


OF 


KEY- 


KEY 


KEY 


WAY 


a 


b 


C 


X2 


Yfi 


1 

3 2 


X2 


_3_ 
3 2 


_3__ 

64 


X2 


X* 


X, 


5 A 


.3.. 
3 2 


_3_ 

64 


y 


Xs 


Y« 


% 


5 

3 2 


s 

64 


H 


H 


Y« 


% 


5. 

•32 


6 4 


M 


% 


3 2 


Vs 


_5_ 
3 2 


ft 
64 


Vs 


% 


_3._ 
3 2 


Vs 


7 
3 2 


7 
6 4" 


Vs 


H 


Vh 


1 


X, 


_3._ 

3 2 




7 


7 


1 


3 2 


64 


1 


M 


y 8 



CENTER OF 
STOCK, FROM 

WHICH KEY 
IS MADE, TO 
TOP OF KEY 



64 
_3_ 
64 
_3_. 

64 



16 

Xe 

<6 

X 

16 
16 

X, 

16 





DIAM- 


THICK- 


NO. OF 


ETER 


NESS 


KEY 


OF 


OF 




KEY 


KEY 




a 


b 


B 


I 


X, 


16 


iVs 


X, 


17 


iVs 


7 
3 2 


18 


i l A 


M 


C 


iVs 


%, 


IQ 


i}4 


X, 


20 


iM 


_1_ 
3 2 


21 


iM 


Va 


D 


iM 


X, 


E 


iM 


% 


22 


1% 


H 


23 


1% 


X* 


F 


iVs 


y 


24 


1X2 


Xa 


25 


1X2 


%, 


G 


1X2 


y 



DEPTH 
OF 
KEY- 
WAY 



ft_ 

3 2 
_£_ 

3 2 

J7__ 
64 

Xs 

32 

.i. 

32 
64 



"3 2 

_S. 
3 2 

1% 



3 2 



CENTER OF 
STOCK, FROM 

WHICH KEY 
IS MADE, TO 
TOP OF KEY 



Ye 



\ 

A 

_5_ 
64 
.5.. 
64 
_£_ 
64 
_ft 
64 
_5_ 
64 
_£_ 
64 
ft_ 
6 4 
_ft_ 
6 4 
_3_ 
3 2 
_3_ 
3 2 

A 

A 

64 



Table 20 — Woodruff Standard Keys 



GEAR PROPORTIONS 



T 33 





ffl S(j 
25 ° 



26 
27 
28 
29 
30 





en 

Cxi ^ 


ft X 
O * 




3 h 


ft w 


H C 

Q 


W ° 

H 




a 


b 


C 


2V* 


3 A 


_3._ 

3 2 


2 l A 


v± 


H 


2 l A 


% 


ft 
32 


2H 


H 


% 


3 l A 


A 


% 






:* O >h 

5 H W 

§<° 

3 ^ f^ 

6 en O 

^ m H 



12. 

3 2 
IX 
32 
11 
3 2 
11 
3 2 

% 



O 


tf >H 






% 


H 


S w 


H W 


< 


B <! 


s « 


w w 


w £ 


H i-l 


§ 


a 


H >< 


P Ph 


& Ph 


< * 


H fc 


ft W 


^ 


535 O 


S 

Q 


W ° 
H 


S« 


e 




a 


6 


c 


_3_ 
3 2 


31 


3^ 


% 


7 
3 2 ■ 


_3_ 

32 


32 


3^ 


^ 


H 


3 2 


33 


M 


% 


9 
3 2 


3 2 


34 


3^ 


b A 


^6 


% 











ph s *, 2 *, 

g o W H « 

« ^ W r 

H M W ^ O 

£ y w a « 

y O £ Ph 

U H S rn P 



W 






% 
% 
% 
% 



Table 21 — Woodruff Special Keys 



DIAMETER 


NUMBER OF 


DIAMETER 


NUMBER OF 


DIAMETER 


NUMBER OF 


OF SHAFT 


KEYS 


OF SHAFT 


KEYS 


OF SHAFT 


KEYS 


%rVs 


I 


%-% 


6, 8, 10 


i^-iJTe 


14, 17, 20 


7 X,-V2 


2, 4 


I 


9, ". 13 


I,^-I 5 /8 


15, 18, 21, 24 


%-Vs 


3, 5 


lJ6-lH 


9, 11, 13, 16 


i%-iM 


l8, 21, 24 


%-u 


3, 5, 7 


I« 


11, 13, 16 


I%"2 


23, 25 


% 


6, 8 


iM-i% 


12, 14, 17, 20 


2^6-2^ 


25 



Woodruff Standard Keys 10 Use with Various Diameter Shafts. 



For heavy work the ordinary key will not answer: it is often necessary to 
put in two keys diametrically opposite, and for extremely heavy work, what 
is known as the "Kennedy" key, Fig. 89, is used. This is the only key that 
will answer the requirements of rolling-mill work. At the armor-plate mill 
of the Carnegie plant this type of key is used in the 2 2 -inch shaft of rolls that 
reverse on an average of 20 times per minute. No other key would stand up 
to this work. These keys are made approximately one-quarter of the shaft 
diameter, and located in the gear so that the corners of the keys intersect 
the bore. It is not necessary for the bottoms of the keys to be on a vertical line. 
The keys are made to a taper of A inch per foot on the top for a driving fit, 



134 



AMERICAN MACHINIST GEAR BOOK 



sides of Key a neat Fit the sides being just a neat fit. The 

Top of Key Tapers % Inch per Foot /,-.-, ,» r i 

shaft is first bored for a press nt, then 
rebored about yU of the shaft diameter 
off center; the keyways are cut in the 
eccentric side. That portion of the bore 
opposite the keys remains as originally 
bored to within one-tenth of the shaft 
diameter below the center line. 

The "Kennedy" key is especially desir- 
able where it is necessary to move the 
gear for any considerable distance on the 
shaft before securing. 

For street-railway work the gear is 
often pressed on the axle, not using a 
key of any description. 
Where a sliding gear is used for heavy work, three keys, having radial sides 
as illustrated by Fig. 90, are generally employed. An example of this is the 




FIG. 89. THE KENNEDY KEY. 





FIG. 90. KEYWAYS FOR HEAVY SLIDING GEAR. FIG. 91. FOUR-KEYED SLIDING GEAR. 

gear drive for vertical rolls. This style of key has a distinct advantage over 
the four-key type with parallel sides as illustrated by Fig. 91. 



BORE 

The bore of a gear is supposed to be standard, any allowance for a fit being 
made in the shaft. This follows out the practice of the manufacturers of 
cold-rolled shafting, that is, to make the shaft enough under size for a slid- 
ing fit in a gear or pulley which is bored standard. 



GEAR PROPORTIONS 135 

An exception to this is when press or shrink fits are desired. In this case 
the allowance is made in the gear, the shaft being turned standard. 

PRESSURES AND ALLOWANCES FOR FORCE FITS 

The Lane & Bodley Company of Cincinnati, O., furnishes the following 
data. For several years this firm has been keeping a record of observations 
on press fits with a view to making an analysis of them when a sufficient body 
of data had been accumulated, and thus obtaining a guide for future practice. 
Hundreds of cases of such press fits have been recorded, forming a body of 
data which is probably unequaled. See Chart 9. 

MEASUREMENTS AND PRESSURE READINGS 

In these records the measurements have been made with great thoroughness. 
Both plug and hole have been measured on two diameters and at both ends, 
the average of these micrometer readings being taken as the true diameter. 
The pressures have been read at the beginning, middle, and end of the length 
of the fit; the material of both plug and ring, the length of the fit, the radial 
thickness of the hub, the areas and volumes of the fitted surfaces, and 
some other minor points have been noted, 24 entries being regularly made 
for each case. The resulting chart will thus be seen to have a very broad 
foundation. 

In ordinary cases the quantities which are fixed by the conditions are the 
nominal diameter and length of the fit, the radial thickness of the hub, and the 
material. With these given it is required to find the press allowance for a 
given pressure to force the plug home. 

HOW THE PRESSURE VARIES 

Regarding the influence of these various factors, the pressure varies: 

1. Directly as the surface of the fit for a given diameter. 

2. Directly as the press-fit allowance, this allowance being such as not 
to stretch the metal beyond the elastic limit. 

3. As some function of the radial thickness of the hub, which, while not 
determined mathematically, is shown in the chart. 

4. With the materials used in a manner not yet determined owing to in- 
sufficient data. The chart is for steel or iron shafts and cast-iron cranks. 
Cast-steel cranks require much heavier pressures for the same press-fit al j 
lowances, but how much heavier cannot at present be said. 

5. With other conditions, which cannot be formulated, and which lead to 
erratic results in the observations make it impossible to formulate a rule 
or construct a chart which shall give other than approximate results. 




20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 

Pounds 

CHART 9. FORCE-PIT RELATIONS SHOWING LANE & BODLEY PRACTICE, 

I36 



GEAR PROPORTIONS 137 

VARYING CONDITIONS 

Among these are the nature of the surfaces as regards smoothness, the 
varying character of materials going under the same name, the shape of the 
crank and the speed with which the work is done. The term cast iron in- 
cludes materials of widely varying hardness and other properties, and it is 
apparent that the web of a disk crank would have an influence not expressed 
by the radius of the hub. If a counterbalance were cast in the disk, this and 
the crank arm would naturally produce an effect on the effective thickness of 
the hub which would be different from the effect of the arm alone on a plain 
crank. Again, with a plain crank, the arm, being taper, would reinforce a 
greater arc of the pin eye than of the shaft eye. 

STILL ANOTHER FACTOR 

Another factor, which no doubt introduces some of the discrepancies of 
the diagram, is that while most of the shafts were of steel, some were of wrought 
iron, and no discrimination between these materials has been made in the 
analysis. These considerations explain the erratic results obtained, but it 
is nevertheless plain that the observations follow the general direction of 
the curves in a very marked manner. 

AVERAGES SHOWN BY THE DIAGRAM 

The plotted observations, it should be remarked, are in most cases the 
averages of many. The figures attached give the percentage of radial hub 
thickness to plug diameter. It will be observed in this connection that the 
discrepancies grow less as the diameters increase. This is doubtless chiefly 
due to the fact that the percentage of error is always greater with small 
experiments than with large. Its effect is to give increased value to the 
diagram when used with large sizes where it is most needed. 

THE PROBLEM SOLVED BY THE CURVES 

Of course, the holding power of these fits is the real thing desired, but it is 
obvious that the probability of adequate experiments being undertaken to 
determine this in large sizes is slight. The problem as it presents itself in 
the shop calls for the determination of the press-fit allowance to give a re- 
quired pressure in forcing the parts home, and this the present diagram solves 
with a degree of accuracy sufficient for most purposes. 

In order to reduce the size of the diagram, the portion applying to diameters 
above 13 inches has been detached and placed at the right. For these large 



138 AMERICAN MACHINIST GEAR BOOK 

sizes the observations are too few to justify the drawing of more than one 
curve. The lubricant used in all cases was linseed oil. 

DIRECTIONS FOR USE 

Select the curve which gives the ratio of the radial thickness of the hub 
divided by the diameter of the plug. Below the point of intersection of the 
plug diameter line with the selected curve, read pounds. Multiply this read- 
ing by the area of the fitted surface in square inches, and by the number of 
thousandths of an inch allowed for the press fit. The result will be the pres- 
sure in pounds carried to force the plug home. 

AN EXAMPLE 

The following example illustrates the use of the diagram: Diameter of 
plug, 8 inches, length of fit, 6 inches, diameter of hub, 16 inches, press-fit 
allowance, 0.020 inch. Required, the pressure to force the parts together. 

Radial thickness of hub = 4 inches 

-. r 3 = O.K. 

Diameter of plug = 8 inches 

Finding a point on the 50 per cent, curve opposite 8 inches, and tracing 
downward, we find 52 pounds. Area of fitted surface = 8 X 3.1416 X 6 = 
150 square inches, 52 X 150 X 20 = 156,000 pounds = 78 tons. 



SECTION V 



Bevel Gears 



Cut bevel gears may be divided into three classes: First, that in which the 
teeth are of uniform depth throughout their entire length and of exact 
profile; second, that in which the depth of teeth correctly decreases toward 
the apex, but in which the teeth are originally cut with a correct profile at 
one point only, necessitating subsequent work on the teeth; and third, that 
in which the true taper type of teeth with correct profile at all points is gen- 
erated by a process reproducing the rolling action of a pair of gears in mesh. 
These three classes may then be referred to according to the form of tooth 
cut: first, a modification of the correct 
tooth; second, an approximation of the 
correct tooth; and third, the correct 
form of tooth. 

The principal relationships of the 
various dimensions, angles, etc., of 
standard bevel gears of true form, in 
accordance with the notations and 
formulas on the succeeding pages, are 
as follows: 

For gears meshing at right angles, 
the tangent of the center angle of a 
bevel gear is found by dividing the number of teeth in the gear by the number 
of teeth in the pinion; obviously the complement of the center angle of the 
gear is the center angle of the pinion. If the axes of the gears do not meet 
at right angles more complicated calculations are necessary. See the suc- 
ceeding formulas. 

The face angle is found by adding the angle increment, the angle between 
the pitch line and the face of the tooth, to the center angle. 

The back angle is the complement of the center angle. See Fig. 92. 

The cutting angle is found by subtracting the angle decrement, the angle 
between the pitch line and the bottom plane of the tooth, from the center 
angle. 

139 




FIG. 92. 



90" 
Face Angle Back Angle 

LOCATION OF FA.CE AND BACX 
AMGLES. 



140 



AMERICAN MACHINIST GEAR BOOK 




FIG. 93. DIAGRAM AND NOTATION FOR BEVEL GEAR. 



BEVEL GEARS 141 

When bevel gears are to be milled with parallel depth teeth, the back 
angle becomes the complement of the face angle, and the cutting angle is 
obtained by subtracting the angle increment instead of the angle decrement 
from the face angle, making the face angle of the gear the same as the cutting 
angle of the pinion and making the clearance the same at both ends of the 
teeth. The back and center angles are the same. 

The tangent of the angle decrement is found by dividing twice the sine of 
the center angle by the number of teeth. This holds true, however, only 

for gears having an addendum of - or 0.3183 p r (Brown & Sharpe standard). 
When special teeth are used, the angle increment equals the addendum 
divided by the apex distance (- ) , and the angle decrement equals the adden- 
dum plus the clearance divided by the apex distance ( J . When the 

addendum equals |, the angle increment is found by dividing the produet 

of 2.314 and the sine of the center angle by the number of teeth. 

The diameter increment is found by multiplying the addendum by the 
cosine of the center angle. Twice this diameter increment added to the 
pitch diameter of a bevel gear gives its outside diameter. 

The apex distance is found by dividing the pitch diameter by twice the 
sine of the center angle and is the same for both gears of a pair. 

When turning gears in large quantities, the work can be carried forward 
more expeditiously if the length of face is given measured parallel with the 
center line of the bore. This distance R is found by multiplying the face 
by the cosine of the face angle. 

Time may also be saved in turning bevel gears if the depth of rim at the 
front end is given. This will allow the workman to finish the face, front end, 
and bore of the gear during the first operation, while it is held in the chuck 
by the back hub. Ordinarily the depth is made 0.2 inch per inch circular 
pitch, but this can be varied to suit requirements, provided the rim depth 
is sufficient to allow for a full tooth at the small end. See Fig. 94. 

The pitch diameter at the small end of the teeth is found by subtracting 
twice the product of the face and the sine of the center angle from the pitch 
diameter at the large end of the gear. 

The outside diameter at the small end bears a similar relationship to the 
outside diameter at the large end. 

The number of teeth for which the cutter should be selected is equal to 
the number of teeth in the gear or pinion divided by the cosine of its center 
angle. 



142 



AMERICAN MACHINIST GEAR BOOK 



The thickness of the tooth at the small end is found by dividing the prod- 
uct of the thickness of the tooth at the large end and the difference between 

the apex distance and the face by the apex distance. Or: 

/ (a-b) 



a :t :: a — b : /'. Therefore t' = 



a 



All dimensions for the small end of the tooth may be determined in this 
proportion. 




DEPTH OF FRONT OF RIM. 



NOTATION AND FORMULAS FOR MITRE GEARS 



Center angle. 
Face angle. 
Cutting angle. 

Angle increment. 



E = 45 degrees. 
F = 45 deg. + /. 
C=45deg. --K. 

Tan. J = 



Angle decrement. Tan. K = 

Diameter increment 
Backing. 
Outside diameter. 

Apex distance; 

Distance from apex to point of tooth — 
large end. Parallel with axis. 



s 

— , or 
a 

s+f 

= S or 

a 

V = s 0.70711. 

Y = s 0.70711. 

D = d+2V. 
d 



1.414 1 

—^7 — when s = — • 

N p 

1.636 , 1 



N 



when s = — « 
P 



a = 



0.1414 



p= i 



, or d 0.70711. 



F. 



BEVEL GEARS 



143 



Face, measured parallel with axis. 

Depth of rim at small end. 

Pitch diameter at small end. 
Outside diameter at small end. 
Number of teeth for which cutter should 
be selected. 

Thickness of tooth at small end. 



R = b cosine F. 

-. W+ 0.2 p'(a-b) 

M = — 0.70711. 

a 

d sf = d — 2 (b 0.70711). 

iy= D - 2 (bsineF). 

N 




E- 

F 
C 
J 

K ■■ 
V 
Y 
d- 

D ■■ 

P'- 

P- 
d s - 

D*- 

t ■■ 
t'- 

a ■■ 
H 



FIG. 95. MITRE GEAR. 

NOTATION FOR BEVEL GEARS 
AXES AT 90 DEGREES 

center angle. 

face angle. 

cutting angle. 

angle increment. 

angle decrement. 

diameter increment. 

backing, or distance from point of tooth to pitch line. 

pitch diameter. 

outside diameter. 

circular pitch. 

diametral pitch. 

pitch diameter at small end. 

outside diameter at small end. 

thickness of tooth at largest pitch diameter. 

thickness of tooth at smallest pitch diameter. 

apex distance. 

distance from pitch line to apex. 



144 



AMERICAN MACHINIST GEAR BOOK 



GEAR 


PINION 


REMARKS 










REQ. 


FORMULA 


REQ. 


FORMULA 




Ei 


Ni 
Tan. Ei = ^ 


E t 


90 -Ei 




Fi 


Ei + / 


F 1 


Ei + J 


Subtract from 90 
for use in lathe. 


C 2 


Ei -K 


Cx 


Ei -K 




/ 


2 sine Ei 
Tan. J = -\r ' 

iv 2 


/ 


2 sine Ei 
Tan. J = Ni 


Same for both gear 
and pinion. 


K 


2.314 sine Ei 
Tan.K- ^ 


K 


2.314 sine Ei 
Tan. K = ^ 


Same for both gear 
and pinion. 


Vl 


s cos. Ei 


Vi 


•S 1 cos. Ei 


Vi or pinion same 
as Yi for gear. 


F 2 


s sine Ei 


Y 1 


S sine Ei 


Fi for pinion same 
as Vi for gear. 


D 2 


di + 2V1 


A 


di + 2F1 




a 


di 


a 


di 


Same per both 




2 sine Ei 


2 sine Ei 


gear and pinion. 


Pi 


d, 
— -Yi 


Pi 


4_ Fl 

2 




R* 


b cos. Fi 


Ri 


b cos. Fi 




Hi 


2 


Hi 


di 

2 




Mi 


W + 0.2 p' (a-b) 

sine Ei 


Mi 


W + 0.2 p' (a-b) 

sine E\ 
a 




di s 


di — 2 (b sine Ei) 


d? 


di — 2 (b sine Ei) 




D2 3 


Di — 2 (b sine F2) 


D,* 


Di — 2 (b sine F\) 




Cutter 2 


Ni 
cos. Ei 


Cutteri 


Ni 
cos. Ei 




f 


t(a -b) 
a 






Same for both gear 
and pinion. 



Formulas for Bevel Gears, Shafts at 90 Degrees 



BEVEL GEARS 



45 



P = distance from point of tooth to axis of mating gear. 
R = face measured parallel with axis. 
M = depth of rim at front end. 
b = face. 
5 = addendum. 
W f = whole depth. 
N = number of teeth. 

FORMULAS FOR BEVEL GEARS AT ANY ANGLE 

Many formulas have heretofore been published for determining the angles 
and dimensions of bevel gears not at right angles, all of which are more or less 
confusing. It is simply a matter of rinding the center angles; all other angles 
and dimensions are obtained as for ordinary bevel gears at right angles, each 
gear being figured separately. This also applies to the use of tables, the table 
being entered for each gear separately, according to its center angle and in- 
dependent of its mate. 

When the axes are not at right angles there are four other combinations; 
axes less than 90 degrees, see Fig. 96; axes greater than 90 degrees, Fig. 97; 
crown bevel gears, Fig. 98; and internal bevel gears, Fig. 99. The center 
angles for these gears are found as follows: 

N 2 = number of teeth in gear. 
Ni = number of teeth in pinion. 




FIG. 96. BEVEL GEARS WITH AXES LESS 
THAN 90 DEGREES. 



American ^fackinUt 
FIG. 97. BEVEL GEARS WITH AXES GREATER 
THAN 90 DEGREES. 



Tan. E = 



sine B 



N x 



Tan. E = 



N 2 
E'= B- E. 



-f- cos. B 



^L 

N, 



sine (180 — B) 

— • 

— cos. (180 — B) 



E'= B - E. 



146 



AMERICAN MACHINIST GEAR BOOK 




American Machinist 
FIG. 98. CROWN BEVEL GEARS. 



FIG. 99. INTERNAL BEVEL GEARS. 

sine B 



Tan. E = 



E = 90 . 
E'= B - 90. 



sine B — 

N 2 



E'= E- B. 



USE OF BEVEL GEAR TABLE 

To use the bevel gear table, Table 23, divide the number of teeth in 
pinion by the number of teeth in gear; the quotient will equal the tangent of 
center angles. Find the nearest number in the table of tangents and on the 
same line on the left will be found the degrees, and at the top of the column 
hundredth degrees for the center angle of pinion. On the same line of the 
right will be found the degrees, and at the bottom of the column hundredth 
degrees for gear. 

On the same line on the left will be found the angle increase, which, when 
divided by the number of teeth in the pinion, will give as a quotient the angle 
increase for either wheel. 

To obtain the face angles add the angle increase to the center angle, and to 
obtain the cutting angle subtract the angle decrement from the center angle. 
Now on the same line on the left will be found the diameter increase for the 
pinion, and on the same line on the right will be found the diameter increase 
for the gear. These when divided by the required diametral pitch, equal the 
diametral increase for that pitch, which, added to the pitch diameters, give 
the outside diameters. 



AN ILLUSTRATIVE EXAMPLE 



In a pair of bevel gears, 24 and 72 teeth, 8 diametral pitch; 24 divided by 
72 = 0.3333, which is the tangent of the center angles. The nearest tangent 
in the table is 0.3346, which gives: 



BEVEL GEARS 147 

Center angle of pinion, 18.50 degrees. 
Center angle of gear, 71.50 degrees. 

On the same line at the left will be found the angle increase, 36, which divided 

.... . 36 

by the number of teeth in the pinion will give the angle increase-^— = 1.5 

degrees. This angle added to the center angle will give the face angle. 

The cutting angle is found by subtracting this angle, plus 16 per cent., from 
the center angle, which in this example would be 1.50 X 0.16 = 0.24, there- 
fore the angle increment would be 1.50 + 0.24 = 1.74 degrees. 

Face angle of pinion = 18.50 + 1.50 = 20.00 degrees. 
Cutting angle of pinion = 18.50 — 1.74 = 16.76 degrees. 
Face angle of gear = 71.50 + 1.50 = 73.00 degrees. 

Cutting angle of gear = 71.50 — 1.74 = 69.76 degrees. 

On the same line to the left will be found the diameter increase for the pinion, 

1.90, which divided by the pitch, or — '--— , = 0.237 inches. On the right of this 

8 

same line will be found the diameter increase for the gear, 0.65, which divided 
by the pitch, or — '— — , = 0.081 inches. 

o 

The pitch diameter of the pinion is — - = 3 inches. The pitch diameter 

o 
72 

of the gear is -*=- = 9 inches, 
o 

Outside diameter of gear = 9 + 0.081 = 9.081 inches. 
Outside diameter of pinion = 3 + 0.237 = 3.237 inches. 

For gears not at right angles, first determine the center angle, and enter the 
table for each gear separately. 



148 



AMERICAN MACHINIST GEAR BOOK 



Number of Teeth in Wheel 



13 



14 



15 
16 

17 
18 



ig 



23 



24 



25 
26 
27 
28 



29 



3° 



31 



32 



33 



34 



35 

36 
37 
38 



39 



40 



41 



42 



42 



2° 38 

3° 2' 
2° 37 
3° 1' 
2°3S 
2° 59 
2° 35 
2 58 

2° 34 
2° 57 
2 ° 33 

2° 56 
2° 31 

2° 54 

2° 30 
2° 52 
2° 29 

2° 50 
2° 28 

2° 49 

2° 26 
2° 48 
2° 25 
2° 46 
2° 24 

2° 44 

2° 22 

2° 43 

2° 20 
2° 41 
2° 19 

2° 39 
2 o J 7 
2° 37 

2° 16 
2° 36 

2° 14 

2° 34 

2° 13 
2° 32 
2° II 
2° 31 
2° IO 
2° 29 

*y 

2° 27 

2 o7' 

2° 25 
2° 6' 
2° 24 

2 o5' 

2° 22 

2° 20 
2° I' 
2° 18 

*> 
2° 17 

i°58 

2° 15 

i° 57 
2° 13 



41 



2" 42' 
3° 6' 

2° 40' 

3° 5' 

2° 38' 

3° 4' 

2° 38' 

3° 2' 

2° 37' 

3° 1' 

2 ° 3s ; 
2 59 

2° 33' 
2° 58' 
2° 32' 
2° 56' 
2° 30' 
2° 54' 
2° 28' 
2° 52' 
2° 27' 

2° 51' 
2° 26' 
2° 49' 
2° 24' 
2° 47' 
2° 23' 
2° 46' 
2° 22' 
2° 44' 
2 19' 
2° 4 2' 
2° 18' 
2° 40' 
2° 17' 
2° 38' 
2° IS' 
2° 36' 
2° 13' 
2° 34' 
2° 12' 

2° 33' 
2° II' 

2° 31' 
2° 8' 
2° 29' 

2 o 7 ' 
2° 27' 

2° 6' 

2° 26' 

2 >' 
2° 24' 

2° 2' 

2° 22' 

2°i' 

2° 20' 

i°59' 

2° 18' 

i°58' 
2 16' 



40 



2" 45' 
3° 10' 

2° 44' 

3 >, 

2 43 

3° 7' 

2° 4 l' 

3° 6' 

2° 40' 

3° 4' 

2° 38' 

3° 3' 
2° 37' 
3° 1' 
2° 35' 

2° 59' 
2° 33' 
2° 58' 
2° 32' 
2° 56' 
2° 30' 
2° 54' 
2° 28' 
2° 52' 
2° 27' 
2° 51' 
2° 26' 
2° 49' 
2 24' 
2° 46' 
2° 22' 
2° 44' 
2° 21' 

2° 43' 
2° 19' 
2° 4 l' 
2° 17' 

2° 39' 
2° 16' 

2° 37' 
2° I 4 ' 

2° 35' 

2° 12' 

2° 33' 
2° II' 

2° 31' 

>y ', 

2 30 
2° 8' 
2° 28' 
2° 6' 
2° 26' 

2 I 4 ', 
2° 24' 

2° 3' 
2° 22' 
2° l' 
2° 20' 



39 



2" 48 
3° 15 
2° 47 
3° 13 
2° 46 
3° 12 
2° 44 
3° 10 
2° 43 
3° 8' 
2° 42 
3° 7' 

2° 40 

<5' 

2° 38 

3° 3' 
37 
3-i' 
2° 35 
2° 59 
2° 33 

2° 58 
2° 32 
2° 56 
2° 30 

2° 28 

2 o 52 
2° 26 

2° 49 

2 24 

O 7 
2° 23 

2° 45 

2° 21 
2° 44 
2 19 
2° 42 
2° 18 

2° 39 
2° 16 

2° 37 
2° 14 
2° 36 
2° 12 

2° 34 
2° II 

2° 32 

*y 

2 30 

2° 8' 
2° 28 
2° 6' 
2° 25 
2° 5' 
2° 23 



38 



53 
19 
52 

3' 18 
' 50 
16 
48 
14 
47 
13 
45 

3" 11 
43 
9' 
42 

3" 7' 
40 

3" 5' 
38 
3" 3' 
2 37 
3° 1' 
2° 35 
59 
2" 33 
2° 58 
2° 31 

2° 56 
2° 29 

2 > 
2 27 

2 o 51 
2° 26 

2° 49 

2 24 

2° 47 

2° 22 

2° 44 
2 20 

2° 42 
2° 18 
2° 40 
2° 16 
2° 38 
2° 14 
2° 36 
2° 13 
2° 34 
2° II 
2° 32 

<9' 

2° 30 
2° 7' 
2° 27 



37 



2" 57 
3° 24 
2° 56 
3° 23 
2° 54 
3° 21 
2° 53 
3° 19 

2 ° 51 
3 17 

2° 49 

*:« 

2 47 

3° 13 
45 
3" 12 

>3 
3 9 

42 

K 7 ' 
2 40 

2° 38 

3° 2' 

2° 36 

3° 1' 

2° 34 

2 °o58 
2 32 

2° 56 
2° 30 

2° 53 

2° 28 

2° 51 
2° 26 

2° 49 

2° 24 
2° 46 
2° 22 
2° 44 
2° 20 

2° 43 

2° 18 
2° 40 
2° 17 

2° 37 

2° 15 

2° 35 
2 13 
2° 34 

2° 12 
2° 32 



36 



3"2' 

3° 29 
3°o' 
3° 28 
2° 58 
3° 26 
2° 56 
3° 24 

2 > 
3 22 
2° 53 
3° 20 
2° 5i 
3 17 
2° 49 
3° 16 
2° 47 
3° 14 
2° 45 
3° 11 
2° 43 
3° 8' 

I 4 , 2 
3° 6' 

°39 

3° 4' 
2 37 
3° l' 
2° 35 
2° 59 
2 33 
2 57 
2° 31 
2° 54 

2° 29 
2° 52 
2° 27 

2° 49 

2 24 

2° 47 

2° 23 

2° 45 

2° 21 
2°*42 
2° 18 
2° 40 
2° 17 
2° 38 
2° 15 
2° 36 



3S 



3° 6' 
3° 35 
3° 4' 
3° 33 
3° 3' 
3° 31 
3° 1' 
3° 29 
2° 58 
3° 26 
2° 57 
3 24 

2 > 
3 22 

2° 53 
3° 19 

2° 50 

3° 17 

2° 48 

So 1 * 
2° 46 

3° 12 

2° 44 

3°io 

2° 42 

O' 

2 40 

3 >' 
2° 38 

3° 2' 

2° 36 

3° o' 
2° 33 

2° 58 
2° 31 

2 54 

2° 29 
2° 52 
2° 27 
2° 50 
2° 25 
2° 48 
2° 23 
2° 45 
2° 21 
2° 42 
2° 19 
2° 40 



34 



3 11 
3° 40 
3° 9' 
3° 38 
3° 7' 
3° 36 
3° 4' 
3° 34 
3° 3' 
3° 31 
3° 1' 
3° 27 

2° 58 

3° 26 
2° 57 
3° 24 
2° 54 
3° 21 
2° 52 

3 > 
2° 49 
3° 16 

2 >7 

3 o I4 

2° 45 
3 11 
2° 43 
3° 8' 

2° 40 

3° 6' 
2 38 
3° 3' 
2° 36 
3°o' 
2° 34 
2° 57 

2° 32 

2 o 55 
2° 29 

2° 53 

2° 27 

2 25 
2° 48 
2° 23 
2° 45 



33 



3" 15 
3° 46 
3° 14 
3° 44 
3° 12 
3° 42 
3° 10 
3° 39 
3° 7' 
3° 37 
3° 5' 
3° 34 
3° 3' 
3° 31 
3°o' 
3° 29 
2° 58 
3° 26 

2° 56 

3° 23 
2° S3 
3° 20 

2° 51 

3° 18 

2° 48 

3 o J 5 

2° 46 
3° 12 
2° 43 
3 9' 
2 41 
3° 6' 
2° 39 

3 o 3 ' 

2° 37 
3° 1' 
2° 34 

2° 58 
2° 32 
2° 56 
2° 29 

2° 53 
2 27 

2° 51 



32 



3 21 
3° 52 
3° 19 
3° SO 
3° 17 
3° 47 
3° 14 
3 45 
3 12 
3° 42 
3° 9' 
3° 39 
3° 7' 
3° 36 
3° 4' 
3 33 
3° 2' 
3° 3i 
3°o' 
3° 27 
2° 57 
3° 24 

2 > 
2 22 

2° 52 
3 > 

2 49 
3° 16 

2° 47 
3° 13 
2° 44 
3° 10 

2° 42 

3° 7' 
2° 39 
3° 5' 
2 36 
3° 1' 
2° 34 

2° 58 
2° 32 
2° 56 



31 



30 



3 27 
3° 59 
3° 24 
3° 56 
3° 22 
3° 54 
3° 19 
3° 5i 
3° 17 
3° 48 

3° 14 
3° 45 
3° 12 
3° 42 
3° 9' 
3° 38 
3° 6' 
3° 35 
3° 3' 
3° 32 
3°o' 
3° 29 

2° 58 

3° 26 
2° 55 
3° 23 

2 > 
3 20 
2° 49 
3° 16 
2° 47 
3° 13 
2° 44 
3° 10 

2° 42 

3° 7' 
2° 39 
3° 4' 
2° 37 

3° i' 



3" 33 
4° 6' 
3° 29 
4° 3 
3° 27 
4 

3" 24 
3° 57 
3° 22 
3° 54 
3° 18 
3° 50 
3° 16 
3° 47 
3° 13 
3° 44 
3° 10 
3° 4° 
3 

3° 36 
3° 4' 
3° 32 
3" 

3" 30 
2 58 
3° 27 

2° 56 

3° 24 
2° 53 
3° 20 

2° 50 

3° 17 
2° 47 
3° 14 
2° 44 
3° 10 

2° 42 

3° 7 



29 



'3° 38 

4° 13 

' 3° 36 

4° 10 

'3° 33 

4° 7' 

'3° 30 

'4°4' 

'3° 27 

' 4° o 

; 3 > 

'3° 56 
'3° 21 
' 3° 53 
'3 
' 3" 49 

:*:** 

3 45 

3° 12 

' 3 > 
3° 8' 

'3° 38 

, 3 o°5 
'3° 34 
'3°2 
'3° 31 
'2° 59 
'3° 28 
'2 56 

X 24 
' 2° 53 
'3° 20 

'2° 51 

'3° 17 
' 2° 47 
;3°I4 



28 



3" 45' 
4° 21' 
3° 43' 
4° 17' 
3° 39' 
4° 13' 
3° 36' 
4° 10' 

3° 33' 
4° 6' 
3° 29' 
4° 2' 
3° 26' 
3° 57' 
3° 23' 
3° 55' 
3° 19' 
3° 51' 
3° 17' 
3° 47' 
3° 13' 
13° 43' 
3° 9' 
3° 39' 
3° 6' 
,3° 35' 
3° 3' 
3° 32' 
3°o' 
3° 28' 
2° 57' 
3° 24' 

2° 53' 

3° 21' 



Table 22 — Addendum and Dedendum Angles for Bevel Gears. Angle Between 



BEVEL GEARS 



149 



Number of Teeth in Wheel 



27 



3" 53 

4° 29' 
3° 49' 
4° 25' 
3° 46 
4° 21 
3° 43 
4° 16' 

3° 39' 
4° 13 
3° 35 
4° 9' 
3° 32 
4° 4' 
3° 28' 
4° 1 
3° 24' 
3° 56' 
3° 21 
3° 53 
3° 17 
3° 

3° 14' 
3 44' 
3° 

3° 39 
3° 7 
3° 36' 
3° 3 
3° 32 
3°o' 
3° 28 



26 





4°o' 




4° 37' 




3° 57' 




4° 33>' 


' 


3° 54' 




4° 29' 




3° 49' 


' 


4° 24' 




3° 45' 




4° 21' 
3° 41' 
4° 16' 




3 U 37' 




4° 10' 
3° 33' 
4° 7' 




3"2 9 ' 

4° 2' 




3 U 2 5 ' 
3° 58' 




3° 22' 




3° 52' 
3° 18' 




3° 49' 


' 


3 :>; 


' 


344 




3 U 10' 




3°4o' 
3° 7' 
3° 36' 



25 



4 7 

' 4° 46 
' 4° 3' 
'4°42 
'3° 59 
'4° 37 
'3° 55 
' 4° 32 
'3° 52 
'4° 27 
'3° 47 
'4° 23 
'3° 43 
'4 18 
'3° 38 

4° 13 
' 3° 34 

4° 8' 
' 3° 30 
'4° 3' 
'3° 26 
'3° 58 
' 3° 22 
'3° 54 
'3° 18 
'3° 49 

i3° 14 
'3° 45 



24 



4° 16 
4° 56 
4° 12 
4° 5i 
4° 7' 
4° 46 
4° 2' 
4° 41 
3° 58 
4° 36 
3° 53 
4° 3° 
3° 49 
4° 25 
3° 44 
4° 19 
3° 40 
4° 14 
3° 35 
4° 8' 
3° 3i 
4° 4' 
3° 27 
3° 59 
3° 22 
3° 54 



23 



4 25 
5° 6' 

4° 20' 
5°o' 
4° 15' 
4° 54' 
4° 9' 
4° 49' 
4° 5' 
4° 44' 
4°°' 
4° 37' 
3° 55' 
4° 32' 
3° SO' 
4° 26' 
3° 45' 
4° 20' 
3° 40' 
4° 14' 
3° 36' 
4° 9' 
3° 32' 
4° 4' 



4" 34' 
5° 16' 
4° 29' 
5° 10' 
4° 24' 
5° 4' 
4° 18' 
4° 58' 
4° 14' 
4° 52' 
4° 8' 
4° 45' 
4° 2' 
4° 39' 
3° 56' 
4° 33' 
3°Si' 
4° 27' 
3° 46' 
4° 21' 
3° 41' 
4° 15' 



4" 43' 
5° 28' 
4° 38' 
5° 21' 
4° 32' 
5° 14' 
4° 26' 

5° 7' 

4° 20' 
5°o' 
4° 13' 
4° 54' 
4° 8' 
4° 47' 
4° 2' 
4° 40' 
3° 56' 
4° 34' 
3° 52' 
4° 27' 



4° 55' 
5° 40' 
4° 48' 
5° 32' 
4° 40' 
5° 25' 
4° 34' 
5° 17' 
4° 28' 
5° 10' 
4° 22' 
5° 2' 
4° 15' 
4° 55' 
4° 8' 
4° 48' 
4° 3' 
4° 4i' 



19 



5° 5' 
5° 53' 
4° 58' 
5° 43' 
4°5o' 
5° 36' 
4° 43' 
5° 27' 
4° 36' 
5° 19' 
4° 30' 
5° 12' 
4° 22' 
5° 3' 
4° 15' 
4° 55' 



5° 16 
6° 6' 
5° 8' 

5° 57 
5°o' 
5° 48 
4° 52 
5° 38 
4° 45' 
5° 29' 
4° 37' 
5° 21' 
4° 29' 
5° 12' 



17 



16 



15 



5° 42' 5° 56' 
6° 36' 6° 52' 
5°32'i5°-45 
6° 24' 6° 39' 



5 22 
6° 11 
5° 12 
6° 2' 
5° 3' 
5° 50 



5 37 
6° 24 
5° 22 
6° 14' 



14 



13 



6° 10' 6° 27' 
7° 9' 17 27 
5° 57'|6° 12' 
6° 54' 7 10 
5° 45' 
6° 43' 



6° 43 

7° 46' 



K= Angle Decrement — "^^j^-""*"— -^ 



J = Angle Inclement 




12 
13 
14 
15 
16 

17 

18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
3i 
32 
33 
34 
35 
36 
37 
38 
39 
40 

4i 
42 



53 



•^ 



Sr* 



Axes 90 , Tooth Proportions Brown & Sharpe Standard. By F. Withers 



i;o 



AMERICAN MACHINIST GEAR BOOK 



Ximiber of Teeth in Wheel 



12 

13 

I 
14 

16 

17 

iS 

19 
20 
21 

22 
23 
24 
23 
26 
27 
2S 
29 
30 
3i 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 



72 


71 


70 


69 


68 


67 66 


: ^ 


64 


63 | 62 


61 


60 


59 


1° 34' 


,i°35' 


l°37 


i°38' 


i° 40 


1 41 1 43 


1 

i°45' 


1° 46' 


i° 48' i° 50' i° 51- 


1° 53' 


i° 55' 


i°49' 


;i u 5o 


i° 52' 


i u 54' 


I" 55 


i° 57' i° 58' 


2°0' 


2° 2' 2° 4' 2° 6' 2° 8' 2° IO' 


2 12 


i° 34' 


ii° 35' 


i° 37' 


i u 38' 


1 40 


i°4i' i° 43' 


i°44' 


i° 46' i° 48' i° 50' i° 5i'ji° 53' 


1 03 


i°49' 


i u 50' 


I" 52' 


i° 53' 


1° 55 


i° 56' 1° 58' 


2 U 0' 


2° 2' 2° 4' 2° 5 ' 2° 7' 12° 9' 


2° 12' 


i" 34' 


X u SS ' 


i°37' 


<& 


1 39 


i°4i' i°43' 


K^' 


i°45' i°47' i° so' i° 50' i° 52' 


i°54' 


i u 4 8 ; 


i°5o' 


i u 5i' 


I u 53' 


I°S5 


i°56'i°58' 


2°o' 


2° I' 2° 3' 2° 5' 2° 7' 2° 9' 


2° II' 


i° 34' 


i°35' 


o 6 ; 


1-38; 


l°39 


i° 40' i° 42' 


x >: 


i°45' i° 47' i° 49' i° 5o' 


1° 5i' 


1° 54' 


i°4S' 


i° 5o' 


i u .1' 


l U 53' 


I U 54 


i° 56' i° 57' 


I U 59' 


2°0' '2° 3' 2° 5' 2° 7' 


2°Q' 


2° II' 


i°33' 


i°33' 


i°36' 


i°37' 


O 8 


i° 40' i° 42' 


l°43' 


£°4S' i°46'i 48'i 5 o' 


i°Si' 


T° -3' 

1 33 


i u 47' 


1 49' 


i° 5i' 


i° 52' 


I U 54 


i° 55' i° 57' 


I u 58' 


2°o' '2° 2' l2°4' l2°6' 


2° 8' 


2° IO' 


i° 33' 


, i° 34' 


i°35' 


i°37' 


i u 3S 


,1 40' 1 41' 


1° 43' 


1 45 1 45 1 48 1 5° i 1 50 


1° 53' 


i°47' 


1 49' 


i° 50' 


1° 51' 


l°53 


1 55' 1 57' 


1° 58' 


20 22 24 26 |2 8 


2° IO' 


l0 33' 


i° 34' 


i°35' 


i u 37' 


l°3S 


i°39' i°4i' 


l u 42' 


1 44' 1 45' 1 47' 1 49'i* 5°' 


1° 52' 


i°47' 


1° 49' 




i° 51' 


I U 53 


1 55 1 50 


i u 58' 


i° 59' 2° i' j2° 3' \2° 5' \2° 7' 


2 U Q' 


i°33' 


I U 34' 


k «; 


i° 36' 


i u 37 


1 39' 140' 


I" 42' 


i°43'i 45'i 4;'i°4S'i o 5o' 


i u 52' 


i° 47' 


1° 48' 


i u 5o' 


i u 5i' 


i u S3 


i° 54' i° 56' 


I 57' 


i°5Q' 


2 1' 2 2 2 4 |2 6 


2° 8' 


i° 32' 


K ** 


i u 34' 


1 36' 


K 37 


i° 38' i° 40' 


l l< 


x : ± 2 \ 


i o 45'i°46'i o 4S'i 49' 


i°5i' 


i u 4&' 


i° 4 8' 


1 49' 


i u 51' 


I u 52 


i° 54' i° 55' 


i u 57' 


l u 59' 


2° 0' 2° 2' 2° 4' \2° 6' 


2° 8' 


I°32' 


*>: 


*>: 


1^35' 


"u 37 


i° 38' i°4o' 


i°4i' 


K^A 


i°44' i°45' i°47'|i°49' 


i°5o' 


i°46' 


i u 47' 


1 49 


1 s° 


i u 52 


i° 53' i°55' 


i u 56' 


i u 58 y 


20 21 23]25 


2° 7' 


I°32' 


1 33' 


i u 34' 


i° 35' 


i u 36 


i°3S'i°39' 


I 40' 


i u 4 2' 


i 44'i°45'l 47',i 48' 


i u 5o' 


i u 46' 


lD 47' 


i°48' 


I 4g' 


z l 51 


1 52' 1 54' 


i u 55' 


i u 57' 


i° 59' 2° l' 


2° 2' 2° 4 ' 


2° 6' 


i° 3i' 


I 33' 


o: 


x >; 


i°36 


1 37' 1 39' 


1 40' 


£ >: 


1 43' 1° 45' 


i 46'i 4 8' 


i u 5o' 


i°45' 


i u 4 6' 


i° 4 8' 


1 49 


1" Si 


i°52'i°54' 


i°55' 


i u 57' 


i 5 8'2°o' 


2° 2' |2° 4' 


2° 5' 


tt 


1 32' 


i°33' 


i"34' 


Os 


1 37' 1 38' 


K^\ 


£ >; 


i° 42' i° 44' 


i°45' i° 47' 


l°49' 


l°4"»' 


i u 4 6' 


i°48' 


1 J.9 


i u 50 


1 5i' 1 53' 


■i u 54' 


■ i u S 6' 


I 58' 2° O' 


2°l' 2° 3' 


2 U S' 


i°3o' 


l°32' 


i° 33* 


i° 34' 


i u 35 


i° 36' 1 38' 


i°39' 


1° 40' 


i°42' i°43' 


i°45'i°46' 


i° 4 8' 


I" 44' 


i°45' 


i u 47' 


i°48' 


i u 50 


i° 51' i° 52' 


i° 54' 


i u 56' 


i°57'i°59' 


2° 0' '2° 2' 


2° 4' 


i° 3o' 


i°3i' 


1 32' 


i u 33' 


i°34 


1 33' 1 37' 


i u 38' 


1 40' 


i°4i'l°43' 


i°44' 1° 45' 


i°47' 


i°44' 


i°45' 


i°46' 


i J ;;' 


l°49 


1 50' 1 52' 


i u 53' 


i°55' 


i°56' i°58' 


i° 59' 2 1' 


2 U 3' 


i°3o' 


i°3i' 


I°32' 


i u 33' 


i°34 


i°35' i°37' 


i° 38' 


i°39' 


i° 40' i° 42' 


i°43' i°45' 


i°46' 


i 44' 


l°4«5' 


i°46' 


1 W 


I AQ 


i° 5o'i°5i' 


l u 53' 


l u 55' 


i°56'i°58' 


1° 59' 2° i' 


2 U 2' 


i° 29' 


*>; 


^^ 


1 33' 


i> 


1 55; 1 36; 


^^ 


^^ 


i° 40' i° 41' 


i°43' i°44' 


i°45' 


i°43' 


l U 44' 


i U 4V 


i- 40' 


1- - 


1 49 1 51 


i° 52' 


I" 54' 


1 00 1 3/ 


i° 58' 2° 0' 


2° I' 


i° 29' 


1° 3o' 


i°3i' 


1 32' 


l°33 


l°34' i°35' 


i° 36' 


1° 38' 


i° 39' i° 4°' i°42' i°44' 


i°45" 


I°42' 


i°43' 


l>5' 


i°46' 


i u 47 


I 48' i° 50' 


i° 5i' 


1° 53' 


i° 54' i° 56' i° 57' i° 59' 


2° i' 


I° 2 8' 


1 29' 




K * 2 ' 


1 33 


i°34' i°35' 


i°35' 


i"37' 


i 38'i o 40'i 4l'i°43' 


l°45' 


I°42' 


i°42' 


i°44' 


i U 45' 


i°47 


i°4S'i°5o' 


i°Si' 


i u 53' 


i° 54' i° 55' i° 56' i° 58' 


2° 0' 


I- 28' 


1 29' 


1 30' 


i u 3i' 


I 32 


i°33' i°35' 


i°35' 


i°37' 


1 38 1 40 1 40 1 42 1 44 


i u 4i' 


i° 42' 


z l^' 


X u 45' 


i°46 


1 47' 1 49' 


I 5°' 


1° 52' 


i°53'i 55'i 56'i 58'i°S9' 


i u 2 7 ' 


i° 28' 


i u 30' 


i J 30' 


i° 32 


1 33' 1 34' 


i° 34' 


i u 36' 


1 38 1 39 1 40 1 41 1 43 


i u 4i' 


I U 42' 


i u 43' 


O' 


i°45 


i°47',i 48' 


i u 5o' 


i u 51' 


i°52' i° 54'i°55'i° 57' i° 58' 


I°2 7 ' 


I°2S' 


1 29' 


I „3o' 


i°3i 


i°32';i 33' 


i° 34' 


i°35' 


i° 37' i° 38' i° 40' i° 40' 


i°42' 


:" -z 


iU ^: 


t>; 


1 44' 


*> 


i° 46' i° 48' 


i°49' 


*:&>: 


i° 5i' i° 53' i° 54' i° 56' 


i°57' 


i° 26' 


i u 2 7 ' 


1 29 


i 30 


1 30 


i°32' i°33' 


1 33 


i u 35' 


i° 36' i° 3S i° 39 i° 40' 


i°4i' 


i° 40' 


x >: 


I > 2 ' 


*>: 


I" 4? 


1° 46' 1 47' 


i°4S' 


^ 5 °! 


i° 51' i° 52' i° 53' i° 55' 


1" 56' 


i° 26' 


I U 2 7 ' 


i° 28' 


I" 2Q' 


i°30 


i° 31' i° 32' 


l°33' 


i u 3 5' 


i° 35' .1° 37' .1° 38' i° 39' 


1° 40' 


i°39' 


I 40' 


i° 42' 


1 43' 


i°44 


i° 45' i° 46' 


i°47' 


i° 49' 


i° 50' i° 52' i° 53' i° 55' 


i u 56' 


I U 2< 


1° 26' 


i° 28' 


I°28' 


1 30 


i° 30' l0 32' 


i° 32' 


i" 34' 


i° 35' i° 36' i° 37' i° 38' 


i° 40' 


1° 39' 


1° 40' 


Oi' 


X >' 


i°43 


i° 44' i° 46' 


i°47' 


i°48' 


l°49' i° 51' 1° 52' i° 54' 


i° 55' 


1° 25' 


1° 26' 


iV 


i u 28' 


1 29 


i°3°' i°3i' 


1 32' 


i°33' 


1 34' i° 35' i°36' i° 38' 


l°39' 


K& 


i°39' 


i u 4 o' 


« 


i°43 


i°44' i°45' 


i° a6' 


i°47' 


i° 48' i° 50' i° 51 i° 53' 


i°54' 


i u 2 4 ' 


*"*!? 


i° 26' 


I 27' 


i° 28 


i° 30' i° 30' 


i° 3i' 


i° 32' 


i°33' i° 35' i° 35' i° 37' 


i° 38' 


1° 38' 


i°39' 


i° 40' 


I . 41 ' 


i°42 


i° 43' i° 45' 


1° 46' 


i°47' 


i° 48' i° 49' i° 50' i c 52' 


i° 53' 


I" 2d' 


I°2S' 


I°2 S ' 


X o 2 7' 


I°28 


i° 29' i° 30' 


i u 3i' 


i° 32' 


i° 33' 1° 34' i° 35' i°36' 


i° 37' 


i°38' 


i u 38' 


i°39' 


I 40 


■> 


i°42' i°44' 


i°45' 


i°46' 


O / O n / O / O / 

i°47 1 48 I 49 1 51 


I°52' 


i° 23' 


i J 24' 


1° 25' 


1° 26' 


I U 2 7 


1° 28' i° 30' 


i° 3i' 


i° 32' 


i°33' i° 33' 1° 34' i° 35' 


i u 37' 


i°37' 


i u 37' 


i u 38' 


!° 39' 


1° 40 


1 41' 1 43' 


i° 44' 


i°45' 


1° 46' 1° 48' i° 49' 1° 50' 


I°5l' 


1 23' 


i°24' 


I°2S J 


I U 26' 


1° 26 


1 27' i° 29' 


i° 3°' 


i° 31' 


i°32' i°33' i° 34' i°35' 


l" 36' 


i°36' 


i° 37' 


i°38' 


K^\ 


1° 40 


i° 41' 1° 42' 


i°43' 


i°44' 


1 45' i° 47' i° 48' i° 49' 


i°5o' 


i° 23' 


I U 2 4 ' 


i u 2 4 ' 


l u ** 


1° 26 


i° 27' i° 28' 


i° 29' 


1 30' 


1 31' i° 32' i° 33' i° 34' 


i u 35' 


i°35' 


1° 36' 


i u 37' 


x"38' 


i°39 


1l° 40' i° 41' 


i°42 


i°43' 


i° 44' i° 46' 


l°47' 


i° 4 8' 


i° 49' 



Table 22 — Continued. Addendum and Dedendum Angles for Be\ t el Ge.\rs 



BEVEL GEARS 



ISI 



Number of Teeth in Wheel 



58 



i°S6' 

2° 14' 

i°56' 

2° 14' 

i°5S' 

2° 13' 

1° 55' 
2° 13' 

i° 55' 
2° 12' 

x >; 

2° 12' 

1° 54' 
2° II' 

2° IO' 

i°53' 
2° 10' 
i° 52' 

*\*'. 

I 51 
2° 8' 

i°5o' 

2 X 

K 7 , 

l° 49 ' 
2° 6' 
l°49' 
2° 5' 
i° 48' 
2° 4' 
l°47' 
2° 3' 
i° 4 6' 

2° 3' 
i°45' 

2° 2' 

l°44' 

2°l' 

l°44' 
2°o' 

l°43' 

1 42 

i°58' 

i°4i' 

i°57' 

i° 40' 

i°56' 

i°4o' 

i°56' 

l°39' 

i°55 

l 3 8' 

i°53' 

i° 3 8' 

i°52' 

i°36' 

i°5i' 

i°36' 

i°So' 



57 



i°58' 

2° l6' 

i°58 
2° 16' 
i°58' 

2° 15' 

i°57' 
2 15' 
i°56' 

2° 14' 

i°56' 

2° 14' 
1° 55' 
2° 13' 

i°55' 

2° 12' 

°54' 

2° 12' 

J 54' 
°ii' 

°53' 
D io' 

:?' 

°52' 
2° 8' 



56 



51' 



2° 8' 

So' 

2" 7' 

i°5o' 

2° 6' 

i°4g' 

°S' 

i° 4 8' 

°4' 

°47' 
2° 3' 
46' 

2"2' 

i°45' 
2' 

1" 45' 

2°o' 

i°44' 

i°59' 

i°43' 

i°58' 

i° 42 

i°57 

i°4i 

i°57 

i° 40 

i° 5 6 

i° 39' 
i°54' 
i° 39' 
i°53' 
i°38' 

I°52' 

i°37' 
i°5i' 



19 

2°o' 

2° 18' 
2°o' 
2° 18' 

2 17 

i°58' 

2° 17' 

1° 57' 
2° 16' 

i°56' 

2° 15' 

i°56' 

2° 14' 

i°55' 

14' 

X o 5S ! 

2° 13' 

i°54' 

2° 12' 

*>' 
2° II' 

2° IO' 
I°52' 

io' 

I" 5l' 

y, 

51 

2° 8' 

x !>' 
2 ! 7 ', 

2 6' 
i°4 9 ' 

2° 5' 

i 4 8' 

a l*' 

i°46' 

2 o 3 ' 
l°46' 



55 



'2' 

5 21' 
3 2' 
' 21' 
5 1' 
' 20' 
'o' 
'19' 
'o' 

•$ 

'18' 

' 59' 

'17' 

'59' 

'16' 

'58' 

' 16' 

'57' 

15' 

'56' 

1 14' 

56' 

13' 

55' 

1 12' 

54' 

1 12' 

S3' 

1 11' 

'53' 

' 10' 

'52' 

'9' 

'5i 

2 8' 

:°5o' 

•°7' 

° 5o' 

!°6' 

:°49' 
'° 5' 



2° 2' 


2° 4' 


r°45' 

2°l' 


i°47' 
2° 3' 


T l 4 ?' 


i° 4 6' 


2 u o' 


2° 2' 


i°44' 


i°45' 


i°59' 


2° i' 


i u 43' 
i°58' 


i u 44' 

2°0' 


X o 42 ', 
1 > 


i u 43' 
i°58' 


i u 4 i' 
i°S6' 


I°42' 

i°57 


i° 40' 
i°55' 


i u 4 i 

x : s6 


i° 39' 


1 40 


i°54' 
i°38' 


i°55 
i° 39 


i u 53' 


i°54 



54 



2° 4' 
2° 24' 

2° 4' 
2° 23' 
2° 3' 
2° 22' 

2° 3' 
2° 21' 
2° 2' 
2° 21' 
2° I' 
2° 2o' 
2°o' 
2° 20' 

°o' 

2 o' 

18' 
I" 59' 

i"S8' 
16' 

1" 57' 
IS' 

i"56' 

2° 14' 

i°55' 

2° 14' 

i°55' 
13' 

12 

!«' 

2° II' 
52' 

io' 

1" Si' 
9' 

1" 5i' 
2 8' 

°5o' 
2° 7' 
i°49' 

°6' 
i°48' 
2°S' 
i°47' 
2° 4' 
i° 4 6' 

2 l*', 
X ° 4 , 5 

2° 2' 

i°44' 

2°0' 

1° 43' 

i°59' 

I° 4 2' 

i°S8' 
i°4i' 
i°57' 
i° 4 o' 
i°S6' 



53 



2 6' 
2° 26' 

2 l 5 ' 

2° 26' 

O' 

2° 25' 
2° 5' 
2° 24' 
2° 4' 
2° 24' 

2° 3' 
2° 23' 
2° 3' 
2° 22' 
°2' 
°2l' 

°l' 

2° 20': 

2°o' 

° 19 ' 

2°o' 
2° 18' 

*>: 

2° 17' 

i°58' 
2 16' 

i°57' 
2° 16' 

2 o 15 

2° I 4 ' 

l0 55 ', 
2° 13' 

X o 54 ' 
2° II' 

x >: 

2° IO' 

2 i 9 ', 

i 8 Si' 

2° 8' 

i° S o' 

*y ' , 

i° 5 o' 
2 6' 
i°49' 
2° 5' 
i°48' 

2 ° 4 ', 
I 8 47' 

2° 3' 
i° 4 6' 

2°l" 

i°45' 

2°o' 

i°44 

X o 59 
l°43 
i°58' 
1° 42 
i°S7 



52 



2" 8' 

2° 2Q' 
2° 8' 
2° 28' 

2° 7' 
2° 28' 
2° 6' 
2° 27' 

2° 5' 
2° 26' 

2° 5' 
2° 25' 

2° 5' 
2° 2 4 ' 

2 >' 
2° 23' 

2° 3' 

2° 23' 

2° 2' 

2° 22' 

2° l' 

2° 2l' 

2°o' 

2° 2o' 

2° o' 

2° ig' 

i°59' 

2° 18' 

i°58' 
2 17' 

i°57' 

2° 16' 

i» S 6' 

2 x * 

X o S < 

2 14' 

2° 13' 

l0 o 5< 
2° II' 

i°S3' 

2° IO' 
I°52' 
2° 9' 

i°5i' 

2° 8' 

1-50' 

2° 7' 

i°4 9 ' 
2 6' 
i° 48' 
2° 5' 

i°47' 

2° 3' 
1° 46' 
2° 2' 

i°45 

2°l' 

i°44 

2°o' 

i°43 
i° 59 



5i 



2" 11' 

2° 32' 
2° lb' 
2° 3l' 
2° IO' 
2° 3°' 
2° 9' 
2° 29' 
2° 8' 
2° 28' 
2° 8' 
2° 2 7 ' 

2° 7' 
2° 26' 
2° 6' 
2° 25' 
2° 5' 
2° 24' 

"4' 
24' 
3' 
2- 23' 

°3' 

°22' 

°2' 
2° 2l' 
2° I' 

°2o' 

°o' 

°I9' 

>; 

2° 18' 
2° 17' 

i°57' 

2° 15' 

i°S6' 

2° 14' 

x >; 
2 13 

i°54' 

2° 12' 

i°53' 

2° II' 
I°52' 
2 « 9 ', 

i°Si' 

2° 8' 

i°5o' 
2° 7' 
i° 49' 

2° 6' 

1° 48' 

2° 4' 

i°47' 
2° 3' 
i°46' 
2 2' 
i°45 

2°l' 

i°44 



50 



2" 15' 

2°3S' 

2° 14' 
2° 34' 
2° 13' 
2° 33' 
2° 12' 
2° 32' 
2° IO' 

2° 31' 
2° io' 
2° 30' 

2 o°9' / 

2° 29' 
2° 8' 
2° 28' 

2° 7' 
2° 27' 
2° 6' 
2° 26' 

2 I 5 ', 
2° 25' 

2° 4' 
2° 24' 

2 o 3 ' 
2° 23' 

2° 2' 

2° 22' 

2°l' 

2° 21' 

2°o' 

2° 2o' 

2°o' 

2° 19' 

x >; 

2° 17' 

i°58' 

2° 16' 

i°57' 
2° 15' 
i°S5' 

2° 14' 

i°55' 

2° 13' 

i°54' 

2° II' 

1° 53' 

2° IO' 
I°52' 
2° 9' 

i°5°' 

2° 8' 

1° 49' 

2° 6' 

1° 48' 

2° 5' 

i°47' 

2° 4' 

1° 46' 

2° 3' 

i°45 



49 



16' 
38' 
15' 
37' 
15' 
36' 
14' 
35' 
13' 
34' 
13' 
3S r 
12' 
32' 
11' 
31' 

2"I0' 

30' 

9' 

29' 



28' 

7' 

26' 

6' 

25' 

5' 

24' 

2 I 4 ' 

2° 23' 

2° 3' 

22' 
2' 
21' 
1' 

2" 19' 

2°o' 

2° 18' 

i°59' 

17' 
I" 58' 

2° 16' 

i°57' 

1 :«; 

2 13 
l0 54' 

2°Il' 
I°S2' 
2° IO' 
1° 5l' 
2° 9' 

i°5°' 
2° 7' 
i° 49' 

2° 6' 

i°48' 
2° 5' 
i°47' 
2° 4' 
i° 46' 

2° 2 



48 



2" 19' 
2° 41' 
2° 18' 
2° 40' 

2° 17' 

2° 39' 
2° 16' 
2° 38' 
2° 15' 
2° 37' 

2 x *' 

2° 36' 
2° 14' 
2° 35' 
2° 13' 
2° 34' 
2° 12' 

2° 33' 
2° II' 

2° 3l' 
2° IO' 
2° 30' 

2 y 

2° 29' 
2° 8' 
2° 28' 

O' 

2° 27' 

2° 6' 
2° 26' 

2° 5' 

2° 24' 

*y, 

2° 23' 

2 ° 3 ' 

2° 22' 
2° 2' 
2° 20' 
2°0' 
2°ig' 

x o 59 ! 

2° 18' 

i°58' 

2° 17' 

i°57' 
2° 15' 
i°S6' 

2° 13' 

x >: 
2 12 

i°53' 

2° II' 
I°52' 
2° 9' 
1° 51' 
2° 8' 

i°5o' 
2° 7' 
i° 49 
2° 5' 
i°47 
2° 4' 



47 



2 21 

2° 44 

2° 21 

2° 43 

2° 20 
2° 42 
2° 19 
2° 41 
2° 18 
2° 40 

2° 39 

2° 16 
2° 38 
2 o IS 

2° 37 

2 14 
2° 36 
2° 13 

2° 35 

2° 12 

2° 33 
2° II 

2° 31 
2° IO 
2° 30 

2 29 

2° 8' 
2° 28 

2 °y 

2° 26 

2 :*' 

2° 25 

2 °j 
2 24 

2 o° 3 ' 
2 22 

2° 2' 
2° 21 
2° I' 
2° 20 
2°o' 
2° 18 

X °S9 

2° 17 

i°58 

2° 15 

i°57 

2° 14 

i°55 

2° 13 

*:* 

2 11 

I°52 
2° 9' 

i°5i 

2° 8' 

i°5o 

2° 6' 

i° 49 
2° 5' 



46 



2 24 

2° 47 
2° 24 
2° 46 
2° 23 
2° 45 
2° 22 

2° 44 

2° 22 

2° 43 
2° 21 
2° 42 
2° 19 
2° 41 
2° 18 
2° 40 
2° 17 

2° 39 
2 o X 5 
2° 37 

2 o X S 
2° 35 
2 13 
2° 34 

2° 12 

2° 33 

2° II 
2° 32 
2°IO 
2° 30 

2 y 

2° 29 
2° 8' 
2° 28 
2° 6' 
2° 26 

2 :*' 

2° 25 
2° 4' 
2° 23 
2° 3' 
2° 22 
2° I' 
2° 20 
2°o' 
2° 19 

i°59 
2 17 

i°58 
2 16 
i°57 

2° 15 

i°55 

2° 13 

X °53 
2 11 

I°52 
2° IO 

i°5i 

2° 8' 

i°So' 

2° 7' 



45 



2" 27 
2° 5i 

2° 27 
2° 50 
2° 26 

2° 49 

2° 25 
2° 48 
2° 24 
2° 46 
2° 23 

2° 45 

2° 21 

2° 44 
2° 20 

2° 43 

2 19 

2° 4 I 

2° 18 

2° 39 
2° 17 

2° 38 
2° 15 
2° 37 
2° 15 
2° 36 
2° 13 

2° 34 
2° 12 
2° 32 
2° II 

2° 31 

2° IO 
2° 30 
2° 8' 
2° 28 

2° 7' 
2 27 

2 : s ' 

2° 25 
2° 4' 

2° 24 

2 °y 

2 22 

2° 2' 
2° 21 
2°l' 
2° 19 

X > 
2° 18 

i°58 

2° 16 

X o 57 
2 14 

*:« 

2 12 

x > 
2 11 

x > 

2 IO 
I°52 
2° 8' 



44 



2 30 
2° 54' 

2° 30' 
2° 53' 
2° 29' 
2° 52' 
2° 28' 

2° SI' 
2° 27' 
2° 50' 
2° 25' 
2° 49' 
2° 25' 
2° 47 ' 
2° 23' 
2° 46' 
2° 22' 

2° 44' 
2°2l' 

2° 43' 
2° 19' 
2° 42' 
2° 

2° 40' 
2° 17' 
2° 39' 
2° 15' 
2° 37' 
2° 14' 

2° 35' 
2° 13' 
2° 34' 
2° 12' 
2° 32' 
2° IO' 
2° 31' 

2 o 9 ' 
2° 30' 

2° 8' 
2° 28' 
2° 6' 
2° 26' 
2° 5' 
2°25' 
r>° a' 

o 4 . 
2 23' 

O' 
2° 21' 

2° 

2 C 20' 

2°o' 

2° I* 

i°58' 

2° l6' 

i°57' 

2° 14' 

i°55' 

2° 13' 

i°54' 

2° II' 

i°53' 

2° IO' 



43 




2° 34' 


12 


2° 58' 




2° 33' 


13 


2° 57' 




2° 32' 


14 


2° 56' 




2° 3l' 


IS 


2" 54' 




2° 29' 


16 


2 U 53' 




2° 28' 


17 


2° 52' 




2° 27' 


18 


2° SO' 




2° 26' 


19 






2 49 




2° 25' 


20 


2° 47' 




2° 24' 


r 2I 


2° 46' 




2° 22' 


22 


2° 45' 




2° 2l' 


23 


2° 43' 




2° 19' 


24 






2 41 




2° 18' 


25 


2° 40' 




2° l6' 


26 


2° 38' 




2" IS' 


27 


2° 36' 




2° 14' 


28 


2° 34' 




2° 12' 


29 


2° 33' 




2° ll' 


30 


2° 32' 




2 U 9' 


31 


2° 30' 




2° 8' 


32 


2° 28' 




2° 7' 


33 


2° 27' 




2° 5' 


34 


2° 25' 




2° 4' 


35 


2° 23' 




2° 3' 


36 


2° 22' 




2° I' 


37 


2° 2o' 




1° 59' 


38 


2° 18' 




1° 58' 


39 


2° 16' 




i° 57' 


40 






2 15' 




i°56' 


4i 


2 l X3 ' 




1 55' 


42 


2 11' 








Angle Between Axes 90 , Tooth Proportions Brown & Sharpe Standard 



1^2 



AMERICAN MACHINIST GEAR BOOK 



AXGLE 


DIAMETER 










IXCREASE. 


INCREASE. 


CEXTER 


CENTER-AXGLE-HUXDREDTH-DEGREES 


CEXTER 


DIAMETER 
IXCREASE. 


DIVIDE 


DIVIDE 


AXGLE 




AXGLE 








DIVLDE 


BY 


BY PITCH 


FOR 




FOR 


BY PITCH 


TEETH IX 


OF 


PIXIOX 


LEFT-HAXD COLUMX READ HERE 


GEAR 


OF GEAR 


PINION 


PIXIOX 












2.00 


O 


.OO 


■ r 7 

.0029; 


■33 
.0058 


.50 
.0087 


.67 

.0116 


•83 
.0145 


1. 00 
•0175 


89 




I 


.OOOO 


•03 


2 


2.00 


I 


•OI75 


.0204 


•0233 


.0262 


.0291 


.0320 


.0349 


88 


.07 


4 


2.00 


2 


•0349 


•0378' 


.0407 


•0437 


.0466 


•0495 


•0524 


87 


.IO 


6 


2.00 


3 


•0524 


•o553; 


.0582 


.0612 


.0641 


.0670 


.0699 


86 


.14 


8 


I.99 


4 


.0699 


.0729 


• 758, 


.0787 


.0816 


.0846 


.0875 


85 


•17 


IO 


I.99 


5 


.0875 


.0904 


•0934; 


.0963 


.0992 


.1022 


.1051 


84 


.21 


12 


I.99 


6 


.IO51 


.10S0 


.1110 


■"39 


.1169 


.1198 


.1228 


83 


•24 


14 


I.98 


7 


.1228 


•1257 


.1278 


■1317 


.1346 


•1376 


.1405 


82 


.28 


16 


I.98 


8 


.1405 


•1435 


.1465 


•1495 


•1524 


.1554 


.1584 


81 


•31 


18 


I.98 


9 


.1584 


.1614 


.1644 


•1673 


.1703 


■1733 


•1763 


80 


•34 


20 


I.97 


10 


•1763 


•1793 


.1823 


•1853 


.1883 


.1914 


.1944 


79 


•38 


22 


I.96 


11 


.1944 


.1974 


.2004 


•2035 


.2065 


•2095 


.2126 


78 


.41 


24 


I.96 


12 


.2126 


•2156 


.2186 


.2217 


.2247 


.2278 


.2309 


77 


•45 


26 


i-95 


13 


.2309 


•2339 


.2370 


.2401 


•243 1 


.2462 


•2493 


76 


.48 


28 


1.94 


14 


•2493 


.2524 


•2555 


.2586 


.2617 


.2648 


.2679 


75 


•5i 


30 


i-93 


15 


.2679 


.2711 


.2742 


•2773 


.2805 


.2836 


.2867 


74 


•55 


32 


1.92 


16 


.2867 


.2899 


.2931 


.2962 


•2994 


.3026 


•3057 


73 


.58 


34 


1.91 


17 


•3057 


.3089 


.3121 


•3153 


.3185 


•3217 


•3249 


72 


.62 


36 


1.90 


18 


•3249 


.3281 


-3314 


•3346 


•3378 


•34i 1 


•3443 


7i 


•65 


37 


1.89 


19 


•3443 


•3476 


•35o8 


•354i 


•3574 


.3607 


.3640 


70 


.68 


39 


1.88 


20 


.3640 


•3 6 73 


.3706 


•3739 


•3772 


.3805 


•3839 


69 


•7i 


4i 


1.86 


21 


•3839 


•3872 


.3906 


•3939 


•3973 


.4006 


.4040 


68 


•75 


43 


1.85 


22 


.4040 


.4074 


.4108. 


.4142 


.4176 


.4210 


•4245 


67 


.78 


45 


1.84 


23 


•4245 


•42 79 


•4314 


.4348 


•4383 


.4417 


•4452 


66 


.81 


47 


1.82 


24 


•4452 


.4487 


•452 2 


•4557 


•4592 


.4628 


.4663 


65 


.84 


49 


1.81 


25 


.4663 


.4699 


•4734 


.477o 


.4806 


.4841 


.4877 


64 


.88 


50 


1.79 


26 


.4877 


•4913 


•495o 


.4986 


.5022 


•5059 


•5095 


63 


.91 


52 


1.78 


27 


•5095 


•5132 


.5169 


.5206 


•5243 


.5280 


•5317 


62 


•93 


54 


1.76 


28 


•5317 


•5354 


•5392 


•5430 


•5467 


•55°5 


•5543 


61 


•97 


56 


i-74 


29 


•5543 


0581 


.5619 


•5658 


.5696 


•5735 


•5774 


60 


1. 00 


57 


i-73 


30 


•5774 


.5812 


0851 


.5890 


•593o 


•5969 


.6009 


59 


1.03 


59 


1. 71 


31 


.6009 


.6048 


.6088 


.6128 


.6168 


.6208 


.6249 


58 


1.05 


61 


1.69 


32 


.6249 


.6289 


•633° 


•6371 


.6412 


•6453 


.6494 


57 


1.08 


63 


1.67 


33 


.6494 


.6536 


.6577 


.6619 


.6661 


.6703 


•6745 


56 


1. 11 


64 


1.65 


34 


•6745 


.6787 


.6830 


.6873 


.6916 


•6959 


.7002 


55 


1. 14 


66 


1.63 


35 


.7002 


.7046 


.7089 


•7133 


.7177 


.7221 


•7265 


54 


1. 17 


68 


1.61 


36 


•7265 


.7310 


•7355 


.7400 


•7445 


.7490 


•7536 


53 


1.20 


69' 


i-59 


37 


■753^ 


.758i 


.7627 


•7673 


.7720 


.7766 


•7813 


52 


1.23 


7i 


i-57 


3* 


.7813 


.7860 


.7907 


•7954 


.8002 


.8050 


.8098 


5i 


1-25 


72 


i-55 


39 


.8098 


.8146 


.8195 


•8243 


.8292 


•8342 


.8391 


50 


1.28 


73 


i-53 


40 


.8391 


.8441 


.8491 


•8541 


.8591 


.8642 


.8693 


49 


131 


75 


i-5i 


4i 


.8693 


.8744 


.8796 


.8847 


.8899 


•8952 


.9004 


48 


i-33 


77 


1.48 


42 


.9004 


•9057 


.9110 


.9163 


.9217 


.9271 


•9325 


47 


1.36 


79 


1.46 


43 


•9325 


•938o 


•9435 


.9490 


•9545 


.9601 


•9657 


46 


i-39 


80 


i-43 


44 


•9657 


•9713 


.9770 


•9827 


.9884 


•9942 


I. OOOO 


45 


1.41 


81 


1.41 


45 


I. OOOO 


1.0058 
"83" ' 


1.0117 


1.0176 

.50 


1-0235 
■33 


1.0295 
•17 


1-0355 

.00 


44 


i-43 




1.00 


.67 










RIGHT-HAXD COLUMX READ HERE 







Table 23 — Diameters axd Axgles of Bevel Gear — Shaft Axgles 90 Degrees 

Becker Milling Machine Company 



BEVEL GEARS 1 53 

MACHINING BEVEL GEARS 

Cutting bevel-gear teeth necessitates quite intricate operations if a correctly 
proportioned tooth is to be finished on other than a generating machine. 
In fact, so intricate would be the required adjustments that quite radical 
departures from a true form of tooth are frequently resorted to — ■modifica- 
tions that are almost considered standard, so universally are they employed. 

The recognized standard tooth for bevel gears, after which all modifications 
are modeled, is of the octoid form. This form of tooth, which was described 
in a preceding section, is simply a modification of the involute and can be 
accurately conjugated only by a generating machine. 

MACHINES FOR CUTTING BEVEL GEARS 

Milling machines are the ones most generally employed in cutting bevel 
gears unless extreme accuracy in the form of teeth is essential. These 
machines readily cut teeth of uniform depth of correct profile, and are also 
used for the cutting operations of bevel gears with teeth of approximately 
correct form — 'the type of bevels which are finished by filing the teeth to 
approximately correct profile. 

Planing machines are employed in similar operations and also for cutting 
teeth which closely resemble the true octoid form. Somewhat special 
planers, known as bevel-gear planing machines, are employed for this more 
exacting work. Three templets are used which guide the simple planing 
tools: a straight-faced one for gashing, and two formed ones, one for each 
side of the tooth space. All the tooth spaces are first gashed, one side of 
each tooth finished, and the gear completed by finishing up the other side of 
the teeth. 

Generating machines are the third type of machines used for cutting the 
teeth of bevel gears. These machines all use the generating principle, but 
either the shaping or milling process may be employed. The generating 
machines operating on the shaping process have a crown gear or its equivalent 
provided with cutting tools representing its sides in successive positions. 
Meshing with this is a large master bevel gear carrying on its arbor the gear 
to be cut. The master and crown gears rotate in mesh, thus presenting the 
work to the cutting tools in successive positions. The blank and the cutting 
tools virtually roll together to conjugate the teeth. Generating machines 
employing the milling process differ in having the sides of the teeth of the 
imaginary crown gear represented by the plane faces of milling cutters. 
The crown gear equivalent is stationary and the master bevel gear is rolled 
over it, thus presenting the work to the cutting tools. The action in the two 
machines is identical, the conjugation of true octoid teeth. 



154 AMERICAN MACHINIST GEAR BOOK 

Of these types of machines for cutting bevel gears, the milling machine is 
generally employed on account of the speed with which the work can be 
performed. Nevertheless the operations required are quite intricate and 
deserve special attention. The meager description of the bevel-gear planer, 
on the other hand, should suffice to explain its operations and, as generating 
machines are fully automatic, little explanation of their action is necessary. 

MILLING BEVEL GEARS 

Though rotary cutters do not lend themselves to the production of really 
high-class bevel gears, in view of the many conditions governing the shape of 
the correctly proportioned tooth, yet gears cut on milling machines are 
nevertheless entirely satisfactory for many purposes, and until the time 
arrives when all gear teeth are really standardized, will continue to be used 
quite extensively. 




a : 

"FIG, IOO. PARALLEL DEPTH BEVEL GEARS. 

The fact that the depth of the true odoid tooth decreases, as well as the 
outside and pitch diameters of a bevel gear, as the cone apex of the gear is 
approached, makes sole reliance upon a rotary cutter impossible. Notwith- 
standing a correct profile to the cutter, the tooth must be gone over with a 
file to remove surplus material, etc. This drawback led to the development 
of the Pentz Parallel Depth Gear in which the depth of the tooth is constant 
throughout its entire length. See Fig. ioo. 

By this radical departure from the true octoid, the necessity of finishing 
the teeth with a file is avoided — the form of tooth being correct at one point, 
it is correct at every other. This form of tooth has one possible drawback, 



BEVEL GEARS 1 5 5 

however, although it is quite generally considered as the equal of the true 
octoid tooth of varying depth. 

The cutter for the parallel depth tooth is selected for the inner or smaller 
pitch circle, so there is always a certain deficiency in the amount of metal at 
the outer ends of the teeth, just as there is a surplus of metal at the inner ends 
of the teeth which has to be filed away when the cutter is selected for the 
outer pitch circle — the method when milling the standard varying depth 
tooth. 

The Pentz Parallel Depth bevel gears are quite distinctive, owing to the 
stubby appearance of the teeth at the outer edge of the gear, but as this form 
of tooth is somewhat easier to mill than the one in which the depth of tooth 
varies, a detailed explanation of the steps required in machining such a gear 
will be more readily grasped than if the more complicated method necessitat- 
ing more adjustments be first considered. The more involved method can 
then be quite easily comprehended. 

The pitch of the cutter is determined from the small end of the gear, and 
the form of the cutter from the average back cone distance, as at b — b (Fig. 
ioo). This will give the average form of tooth, as it is apparent that the true 
form cannot be maintained the entire length of face. The only change in 
form is due to the reduced back cone distance from a to c; there is no change 
due to reducing the depth of cut, as is found in the ordinary methods of 
bevel gear milling, the pitch line of the bevel gear being cut, and the pitch 
line of the cutter coinciding during the entire cutting operation. 

The milling of a parallel or, for that matter, tapered tooth bevel gears 
will be better understood if the pitch is considered at the small end of the 
tooth. When taking side cuts, this pitch alone should be kept in mind and the 
matter will appear in a new light. 

After the first central cut has been taken as illustrated by Fig. 101 the blank 
is rolled to the right, bringing the pitch fine of the left-hand side of space 
parallel with the travel of cutter, as shown by Fig. 102. This movement is 
accomplished by indexing the blank one-quarter as many holes in the index 
plate as are used altogether to space the teeth. The table is then moved 
toward the nose of the spindle a distance equal to one-half the tooth thickness 
at small end, or one-quarter the pitch at small end. This will bring the pitch 
line of gear to the pitch line of the cutter, and the blank in position for first 
side cut, as shown in Fig. 103. 

After this cut has been made, roll the gear blank to the left, one-half as many 
holes in index plate as are used altogether to space the tooth, as per Fig. 104, 
and move the table away from the nose of spindle the thickness of tooth at 
small end; this will bring the other side of the tooth into the same relative 
position, and the blank is in position for the second side cut, as in Fig. 105. 



156 



AMERICAN MACHINIST GEAR BOOK 



It will be noted that the pitch is figured at the small end of the tooth, an 
ordinary spur gear cutter corresponding to the pitch at this point being used. 

The half tone, Fig. 106, shows a pair of parallel tooth gears, sent with W. 
Allen's original article. Their operation was entirely satisfactory and the teeth 




FIG. IOI. FIRST CUT IN MILLING BEVEL GEARS. 



gave no evidence of being filed. These samples were 25 teeth, 10 pitch at 
small end and one-inch face, No. 2, 10 pitch cutter used. 

Referring to Figs. 101 to 105; in moving the table forward and back one- 
half the thickness of space at the small end there is a small error due to the 



BEVEL GEARS 



157 




FIG. I02. POSITION OF BLANK WHEN ROTATED ONE-QUARTER OF THE INDEX. 




FIG. IO3. CUTTER IN POSITION FOR THE FIRST SIDE CUT. 




Distance to 
move Cutter 



FIG. 104. BLANK IN POSITION FOR SECOND SIDE CUT. 



158 



AMERICAN MACHINIST GEAR BOOK 



fact that instead of these moves being made in the direction of the pitch line 
they are made tangent to it, as illustrated by Fig. 107, which is exaggerated 
to show this clearly, c representing the distance actually moved and / the 
theoretical distance. This error, however, is on the safe side, so that setting 
the machine as directed will allow a little clearance, depending, of course, upon 
the diameter and pitch of gear, but will not be noticed except in extreme 
cases. 




FIG. I05. CUTTER IN POSITION FOR SECOND SIDE CUT. 





FIG. 106. A PAIR OF PARALLEL DEPTH BEVEL GEARS IN MESH. 



In milling the taper-tooth type of bevels, very similar steps are taken, 
but, of course, the face and cutting angles are not the same. This naturally 
complicates the holding and adjusting of the work to a considerable extent. 

The cutter should be selected with a correct profile for the outer ends of 
the teeth, but its thickness must be such as to allow it to pass between the 



BEVEL GEARS 



159 



teeth at the inner edge of the gear. The necessary adjustments of the blank 
are made, usually by trial, and the cuts taken in a manner very similar to 
that described for the parallel-depth method. The teeth are milled to a cor- 
rect outer end thickness and the surplus metal toward the inner edge removed 
by a file. See Fig. 108. 

The reason that this excess metal is left by the cutter is that, though the 
set of teeth as cut are correct throughout their entire length at the pitch line, 
their curvature is correct only at the outer end of the teeth. In the octoid 



Filed , Surface 





FIG. 107. DIAGRAM SHOWING ERROR IN 
SETTING OVER CUTTER FOR SIDE CUT 
ON MILLED BEVEL GEARS. 



FIG. 108. THE FILED SURFACE OF 
MILLED BEVEL GEARS. 



form of tooth, for which the cutters were selected, the radius of curvature 
grows less and less as the pitch diameter of the gear decreases — i.e., toward 
its inner edge. Rotary cutters, being unable to alter their curvature, leave 
a surplus of metal outside the pitch line which gradually increases in thick- 
ness as the inner edge of the gear is approached. This surplus can only be 
removed by filing. In performing such an operation, great care is necessary 
not to reduce the thickness of the teeth at the pitch line. Particularly is 
this so if the gear is one of wide face, as the greater the face of the teeth, the 
more filing required. 



THE USE OF GENERATING MACHINES 

If the gears to be finished are stocked out before being mounted on the 
generating machine, these machines are really fully automatic and when 
once set up require no further attention until the job is completed. Stock- 
ing out on generating machines is not to be recommended ordinarily, although 
the larger machines are equipped for this operation. Generating machines 
are essentially finishing machines with delicate adjustment of cutting tools 



160 AMERICAN MACHINIST GEAR BOOK 

that finish a stocked out tooth in one generating operation, so that the 
rougher and heavier work of stocking out should advisably always be per- 
formed on some other and more rugged machine. This will enable the ca- 
pacity of the generating machine to be realized fully, and operations in quan- 
tities carried through rapidly and efficiently. 

EFFICIENCY OF BEVEL GEARS 

The chief cause of decreased efficiency in bevel gearing is due to inaccu- 
racies in the shape and form of the teeth, whereby a lateral thrust is produced 
which tends to force the gears out of mesh. This is particularly noticeable 
in gears finished on milling machines or on planers unless the gears have teeth 
of the parallel-depth type, owing to the fact that it is seldom that the center 
angles of the gear and pinion coincide as they roll together. The tooth 
pressure cannot be normal between all points of engaging teeth and in such 
case a decided lateral thrust tending to separate the teeth is produced. 
Furthermore, the pinion, and usually the gear also, must be overhung so 
that the thrust is greatly increased at the bearing by leverage, materially 
increasing friction, twisting the shaft, etc. 

Even if it were possible to produce a perfect tooth on a generating machine, 
so that the tooth pressure is transmitted normally, inefficiencies that can- 
not be avoided arise, due to the angularity of the gear and pinion shafts. 
The greater the center angle of a bevel gear, the greater, as a rule, the loss 
in efficiency. The most efficient center angle to employ is that of 45 degrees, 
for then both the gear and the pinion suffer equally in respect to angularity. 

Correctly proportioned teeth and adequate shaft diameters with minimum 
backing are particularly essential for the satisfactory operation of bevel 
gears. Unsatisfactory tooth speed and wide differences between the number 
of teeth on the gear and on the pinion are also more serious drawbacks in 
bevel gearing than in spur gearing, owing to the fact that a bevel gear has 
pitch circles and corresponding tooth speeds varying from that of the inner 
or smaller end of the gear through to that of the larger end. 

HARDENING BEVEL GEARS 

The abrasive wear on gear teeth is always appreciable if the profile of the 
teeth varies even slightly from the correct form, and once such deterioration 
commences, it rapidly becomes serious. This is more noticeable in bevel 
gearing than it is in spur gearing, owing to the greater difficulty of securing 
the correct tooth profile in the former. To overcome or minimize this 
destruction, case-hardening of the gears is the logical procedure. 



BEVEL GEARS 161 

The great demand for satisfactory gears, bevels as well as other types of 
toothed wheels, in automobile construction has probably been the main 
reason for the knowledge gained in recent years of the art of case-hardening 
gears. The transmission of the mechanically propelled vehicle must possess 
exceptional wearing qualities to give satisfaction, and, though other intricate 
and delicate machinery also requires gears capable of resisting the abrasive 
action of teeth slipping upon one another, it is the automobile that has made 
this subject one of such general interest. 

Processes of hardening gears, of course, differ in nearly every shop, but the 
general requirements and results secured are similar. The accurately cut 
gear should first be slowly and uniformly preheated to avoid undue distortion 
and to bring the metal to as homogeneous a condition as possible. The gear 
should then be subjected to the chemical treatment by which the skin of the 
metal is prepared for the hardening quench. Finally the quenching opera- 
tion is performed, usually in an oil bath, and the gears rapidly but uniformly 
cooled. 

One process of hardening automobile gears which has proved very success- 
ful and satisfactory is to preheat the gears in a commodious gas forge or oven 
until they become dull red (at a temperature of from 1,300 to 1,400 degrees 
Fahrenheit) and then boil them in a solution of cyanide of potassium for 
about an hour. The gears are then immediately plunged into a bath of 
fish oil which is kept in constant circulation to guard against localized over- 
heating of the oil. 

Gears hardened by this process show remarkably little warping and are 
uniformly hard — 'the case-hardening penetrating an average depth of 0.025 
inch — 'and are practically free from any change in volume or distribution 
of metal. The average shrinkage of the diameter of shaft holes, in a large 
number of automobile transmission gears so treated, is reported to have been 
only about 0.0005 mcn (total shrinkage). 

GRINDING BEVEL GEARS 

Gears that have been most carefully case-hardened usually show some 
deformation, however, and no matter how slight this is their smooth and 
proper action is impossible. Thus the benefits derived from hardening are 
partially counteracted. The distortion that is noticeable in a gear that has 
been carefully hardened, however, is so slight that it can be localized in the 
hub and there corrected. That is, the gear may be mounted on a rotary 
machine so that the teeth run true and any deformity thereby concentrated 
in the hub, which may be rebored, the bore straightened and the hub cor- 
rectly faced. 



1 62 AMERICAN MACHINIST GEAR BOOK 

A simple method of trueing up hardened bevel gears for grinding, and hold- 
ing them in such position, was described in American Machinist, July 20, 
191 1. Briefly summarized, the salient points in this description follow: 
The gears are first clamped to a face plate by a bolt through their bore, set 
true by using a test indicator, and kept in position by four perfectly round 
and straight pins placed equidistant on the periphery of the gear and held 
in place by rubber bands. The gear is then firmly strapped to the face 
plate, the bolt through the bore removed, and the bore and back of the hub 
properly ground. 

Draw-in chucks are also used for holding gears for grinding. The gears 
are held by the roots of the teeth, by the pitch line, or by the top of the teeth. 



SECTION VI 

Worm Gears 

An interesting model of half a dozen sets of worm gears is shown in Fig. 
109. All the gears are of the same diameter with teeth of the same normal 
pitch, though the respective speed ratios of the various sets differ. The hori- 
zontal shaft in making 32 revolutions causes the vertical gear in the fore- 




FIG. 109. MODEL OF SPIRAL GEARS OF VARIOUS RATIOS. 

ground to make but one complete revolution, the second, two complete turns, 
and each of the succeeding vertical gears twice the number of revolutions 
made by the gear immediately in front of it, the furthermost gear making 32 

163 



1 64 AMERICAN MACHINIST GEAR BOOK 

revolutions to 32 turns of the horizontal shaft. The consecutive speed 
ratios, commencing with the set in the foreground, are respectively 32:1, 
16:1, 8:1, 4:1, 2:1 and 1:1, the revolutions of the horizontal shaft being 
named first. 

The first three drives are evidently worms of single, double and quadruple 
thread respectively, but they are also spiral gears as the driven gears are cut 
with spiral teeth, not simply hobbed as is customary in laying out worm 
gears. The balance of the drivers resemble spiral gears even more, par- 
ticularly the most remote, showing that the so-called "worm" is in reality a 
toothed gear of the spiral type. 

NOTATION FOR WORM GEARS 

N = number of teeth in worm wheel. 
n = number of threads in worm. 

p f = circular pitch (distance from center to center of teeth). 
L = lead (advance of worm in one revolution). 
D' = pitch diameter of worm wheel. 
T = throat diameter of worm wheel. 
D = outside diameter of worm wheel. 
F = face of worm wheel. 

a = distance from center line to point of tooth. 

b = length of side. 
d' = pitch diameter of worm. 

d = outside diameter of worm. 
d" — bottom diameter of worm. 

e = radius at throat of worm wheel. 

<j> = angle of sides of face. 

B = center distance. 

R = number of revolutions of worm to one of wheel. 

S = angle of teeth in wheel with axis (used for gashing teeth). 

TT = 3.I416. 

W = working depth. 
W = whole depth. 

/ = clearance. 

/ = thickness of tooth at pitch line. 

t n = normal thickness of tooth at pitch line. 
p' n = normal circular pitch. 

s = addendum. 

U = width of worm thread at top. 
F = width of worm thread at bottom. 
p = diametral pitch. 



WORM GEARS 165 

FORMULAS FOR WORM GEARS 

P 

D' = N p' 0.3183. 
T = (iV + 2) p' 0.3183. 
D = r + 2 (e — e cos. $). 

F = \ 2 ' ^ / ^ d + (0.34 />') , , 

v , or - — - — , when <p = 30 degrees, 

0.5 2 6 

a = F — (b sine cf>). 

b = W + (0.12 p'). 
d' = as small as possible. (See discussion.) 

d = d' + 2 s. 
d" = d - 2 W. 

d' 
e = s. 

2 

F 

<j> = 30 to 35 , or sine <f> = 



B = 



d + (0.34 />')* 



2 



Z>' 7T 7T 

^=o^8^ = ^ 0r ^ = F 
X = £'rc. 

N 
n = R- 

Tang 8 = ~ 

t n 1.15708 , R w , 

t n = t cos. 8, or t = 1 = - — : — when 8 = i4>4 degrees. 

cos. 8 p ° 

f = 0.1 t. (See discussion.) 

. 1.0^36 ,„ .. N 

£7 = 0.335 Pi or — I — * w ee discussion.) 

Y = 0.31 £', or -^— — . (See discussion.) 

p'n 
p'n = p cos , * or p' = -r ^^ 

TF r = 0.6866 p' . (See discussion.) 
Formulas for tooth parts as given for spur gears apply to worm gears. 



i66 



AMERICAN MACHINIST GEAR BOOK 



DISCUSSION OF FORMULAS 

N, D f and p'. The number of teeth, the pitch and the pitch diameter of 
worm gears are calculated in the same manner employed for spur gears. 

T. The throat diameter of a worm gear or wheel corresponds to the out- 
side diameter of a spur gear of the same number of teeth and the same pitch. 

D. The extreme outside diameter of a worm-wheel can be found by the 
formula given, but ordinarily the measurement of a carefully drawn sketch 
is sufficiently accurate. Insufficient stock for a sharp tooth edge, depicted 



fc-a^ 




■ , 4> 1 ->I<- J 0'' 
FIG. IIO. DIAGRAM FOR 
WORM GEAR. 



on Fig. no, is preferable to enough stock for a perfect tooth as it is safer 
to handle a gear without the sharp-cornered teeth and also because such gear 
is of more pleasing appearance. 

F. There is no gain in making greater than 30 degrees, so the second 
formula for F is the one usually employed. 

d f . The angle of the worm, 8, governs in large part the efficiency of a 
worm drive, so the pitch diameter of the worm should be made as small as 

possible. When the lead is fixed, however, d' is also fixed for tang d =— -7/ 



WORM GEARS 



167 



4>. This angle is usually from 30 to 35 degrees, preferably 30 degrees. 
When made as great as 45 degrees and the face of the gear widened to corre- 
spond, the gear will wear out rapidly. 

L. The lead is the advance of the worm thread in one complete revolu- 
tion and is found by multiplying the circular pitch by the number of threads 
in the worm. For instance, a 1.5-inch pitch gear with double threads has 
a 3-inch lead. 

n. The number of threads required for a worm is found by dividing the 
number of teeth in the worm-wheel by the velocity ratio required. For 
example: If a worm gear has 60 teeth and a velocity ratio of 30 to 1 is 

required, the worm should have two threads! — = 2) . 



at pitch 
line 




FIG. III. WORM. 



/, U, Y, and W . The formulas given for these various dimensions apply 
to worms cut to 14^ degrees standard. Any deviation from this standard 
obviously calls for modifications of these formulas, but no fixed rule can be 
advanced. 

The length of the worm (see Fig. in) need be no more than three times the 
circular pitch, as seldom do more than two teeth come in contact with the 
wheel teeth at the same time. It is good practice, however, to make the 
worm about six times as long as the circular pitch so that it may be shifted 
as it becomes worn, the worm nearly invariably wearing more rapidly than 
the wheel. 



i68 



AMERICAN MACHINIST GEAR BOOK 



REVERSIBLE WORM AND GEAR 

The surface of the worm thread constantly slips and slides over the sur- 
faces of the wheel teeth, the direction of slippage following the plane of con- 
tact. The slippage plane being mutually tangent to the face of the thread 
and the face of the wheel teeth, the gliding angle must equal the sum of the 
lead angle and half the angle included between the faces of the worm thread. 
When this gliding angle equals 45 degrees it is evident that it is immaterial 
whether the worm or the gear is the driver, or which is the driven member — ■ 
the worm and the gear will be perfectly reversible in this respect. 

Expressed in the form of a simple equation, the conditions requisite for the 
worm and gear to be perfectly reversible are as follows: 



Y = — = 45 degrees 



where 



Y = lead angle, 

X = angle included between the faces of the worm thread. 

This equation is generahy applicable, but there are reasonable limits to the- 
lead angle that can be efficiently employed. The lower limit for the lead 
angle is about 25 degrees, making necessary an included angle of 40 degrees 
in order to make the gliding angle one of 45 degrees. A more acute lead 
angle would necessitate an included angle so obtuse as not to give adequate 
contact surface. The upper limit is, of course, 45 degrees. In this case there 
could be no included angle. This condition could exist only when the profile 
of the worm thread is perpendicular to the axis of the worm, when the worm 
thread is square or rectangular in cross-sections. 



LEAD 


INCLUDED 


LEAD 


INCLUDED 


JGLE, Y 


ANGLE, 


X 


ANGLE, Y 


ANGLE, X 


25 


40 




36 


18 


26 


38 




37 


16 


27 


36 




38 


14 


28 


34 




39 


12 


29 


32 




40 


IO 


30 


30 




4i 


8 


31 


28 




42 


6 


32 


26 




43 


4 


33 


24 




44 


2 


34 


22 




45 


O 


35 


20 









WORM GEARS 



THE HOB 



169 



The hob for cutting the teeth on the wheel must have an outside diameter 
equal to the outside diameter of the worm plus twice the clearance (see Fig. 
112), but this should advisably be increased by 0.03 X p r to allow for wear. 



W-29V 




FIG. 112. HOB. 



The hob should always be a little longer than the section of the worm 
having any contact with the wheel (see Fig. 113) and, as this depends upon 




PIG. U.S. LENGTH OF THE HOB. 



the diameter of the wheel to be cut, it should be proportioned to the diameter 
of the largest wheel liable to be handled. The correct hob length may be 
found from the following simple formula: 

L = 2\/{D - W) - W 
where 

L = length of hob, 

D = outside diameter of largest gear to be cut, 
W = whole depth of tooth. 



170 



AMERICAN MACHINIST GEAR BOOK 



NUMBER OF FLUTES 

The following article by Oscar J. Beale, which originally appeared in 
American Machinist, June 22, 1899, very ably discusses the proper arrange- 
ment of teeth on a worm-gear hob. 

"In the works of the Brown & Sharpe Manufacturing Company a pair of 
gears was wanted of the spiral or screw type, and it was thought better to make 
the large gear, or member, as a worm and the small member as a worm-wheel. 




FIG. 114. WORM GEAR B AND 
WORM A. 



FIG. Il8. HOB USED FOR CUTTING 
WHEEL B OF FIG. 114. 



Fig. 114 shows the worm and wheel in mesh; A is the worm and B is the 
worm-wheel. The large member, or the worm A, has 43 threads; the lead of 
the worm is 60.3 inches, and the thread pitch, or the axial pitch, is 1.4 inch. 
The small member, or the worm-wheel B, has 7 teeth, and the circular pitch of 
the wheel is, of course, the same as the thread pitch of the worm, 1.4 inch. 

Fig. 115 is an axial section of the worm threads. The threads incline 57 
degrees from the plane perpendicular to the axis, which is so great that, while 
the axial thickness of the thread at the pitch line is T 7 o inch, the actual or the 
normal thickness is not quite tV inch. In Fig. 116 the line C D shows the 
inclination of the threads; C E is the axial pitch, and F G the actual or normal 
pitch. 

In cases where the inclination of the thread is more than 15 degrees, that is, 
in cases where the normal pitch is less than 0.96 of the axial pitch, it is well to 
have the depth and the addendum correspond to the normal pitch. Fig, 117 



WORM GEARS 



l 7 l 



is a normal section of the thread, the depth being the same as a gear tooth of 
equal pitch, which makes the thread look shallow and thick when seen in the 
axial section, Fig. 115. 

The worm-wheel B, Fig. 114, was hobbed, or cut, with the hob shown in Fig. 
118. 

The worm has more than six times as many threads as the worm-wheel; the 
pitch diameter of the worm is four times that of the wheel; the wheel is the 
driver. The hob is made up of a cast-iron body, upon which are fastened lags 
that are arranged in steps in order 
to have the lags alike for con- 
venience in manufacturing. Once a 
large hob was made that did not 
work, because the cutting edges of 
the hob teeth did not trim the tops 
of the worm-wheel teeth narrow 
enough to clear the backs of the hob 
teeth, which jammed so hard that the 
machine could not go. This jamming 
of the backs of the hob teeth upon 

(4 7 »f FIG. Il6. AXIAL AND NORMAL PITCH. 



H-t383-^ 






FIG. 115. 



AXIAL SECTION OF WORM 
THREADS. 



FIG. 117. NORMAL SECTION 
OF WORM THREADS. 



the tops of the worm-wheel teeth was owing in part to incorrect spacing of the 
lags H H, Fig. 118, which will be explained. 

A worm is a screw whose threads have the same outline, upon an axial section, 
as the teeth of a rack, the purpose of a worm being to mesh with a gear. A 
worm gear meshes with a worm. The action of a worm meshing with a worm 
gear is analogous to that of a rack with a spur gear, as stated by Professor 
Willis in his "Principles of Mechanism." In most worms the outlines of the 
threads upon the axial section have straight sides, as in Fig. 115, which cor- 
responds to the sides of rack teeth in the involute system of gearing. 



172 AMERICAN MACHINIST GEAR BOOK 

Fig. 118 is a hob made up of cast-iron body, into which are fastened lags H H. 
Two of these lags are shown detached. The lags were threaded in axial section 
like A, Fig. 115. The resulting teeth were trimmed and backed off, as in the 
detached lag on the left. The numbers in the scale are for inches. 

I have spoken of the failure of a hob because the backs of the hob teeth 
jammed upon the tops of the wheel teeth. This interfering action can be ex- 
plained in several ways; it is analogous to trying to thread a coarse screw in a 
lathe with a tool that does not lead or incline in the same direction that the 
thread inclines. A thread tool inclined for a right-hand thread would soon 
interfere in cutting a left-hand thread. Any grooving tool that has only one 
cutting edge or face must track in the same groove that it cuts. Sometimes 
a tool goes wrong and cuts a groove that bends the tool, which is occasionally 
noticed in cutting off a large piece in a lathe. In cutting a deep narrow groove 
a thin saw sometimes runs so much to one side that the saw is broken. In the 
case of the hob the interfering teeth would neither bend nor break, and so the 
machine had to stop. The teeth of a hob should be so arranged that there will 
be a cutting edge to take off an interfering point as it comes in the way. A 
worm-wheel can be cut with a tool that has only one cutting edge by bringing 
the tool into different positions in relation to the teeth of the wheel. In the 
American Machinist for May 27, 1897, reference was made to the great 
number of cutting edges that a hob must have in order to cut a perfect wheel, 
and a description was given of a machine that cuts worm-wheels with a single 
tool acting in different positions. Such a machine was patented November 
15, 1887, and another July 5, 1898. 

In most hobs the cutting edges are straight, and in consequence the sides of 
the hobbed worm-wheel teeth are made up of straight lines in warped surfaces 
that meet in angles. These angles are often not noticed in worm-wheels of fine 
pitch and in wheels having a large number of teeth; but in wheels of coarse pitch 
and in wheels having few teeth the angles may be quite pronounced. Fig. 119 
shows a worm-wheel that has teeth with hobbly sides on account of these angles. 
This wheel was cut with the hob shown in Fig. 120. The length and the diame- 
ter of the blank were great enough to extend beyond the teeth left by the hob 
that are available to work in connection with the worm. The available part 
of the teeth occupy about two-thirds the length of the wheel through the mid- 
part, as between / and /. Though the teeth are available, yet their sides are 
so hobbly between / and / that they will need to have the angles finished off 
before the wheel can run smoothly with its worm. 

Another kind of stepped action of the hob is seen as grooves near K L, Fig. 
119, which are cut in consequence of the quick travel of the large part K L, in 
proportion to the narrow flats M M, Fig. 120, at the tops of the hob teeth. If 



WORM GEARS 



173 



the travel of K L had been slow enough or if the flats M M had been wide 
enough, there would have been no grooves. 

The circular pitch and the number of teeth of Fig. 119 are the same as in 
J5, Fig. 114. 

It is well known that the cutting edges of a hob must act upon the worm- 
wheel teeth in different positions, and that a tool with a single cutting edge 
must track in a groove cut with itself; 
but it was a surprise to learn that a 
hob of any number of cutting edges 
can be so made that it will absolutely 





FIG. II9. A WORM GEAR OF FEW TEETH 
AND COARSE PITCH. 



FIG. I20. HOB USED IN CUTTING WORM 
GEAR OF FIG. 1 1 9. 



refuse to cut a wheel that has only a few teeth like B, Fig. 114. When the 
workman told me that the hob jammed, I was incredulous, but a glance at 
the work proved that he was right. I could not believe that my previous ex- 
perience had been such that I could have known how to make the hob, yet in 
a few minutes, when the solution came to me, I had the feeling that I must 
have known it well some time in the long past. 

The things that affect this interference might be called variable; there are 
several of these variables. I am unable to give a rule that will indicate the 
conditions in which interference would be objectionable; yet, while limits may 
not be easily defined, an understanding of a few extremes may enable us to keep 
away outside these limits. 

One way of explaining the interference is based upon the fact that, in a gear, 
any point outside the pitch circle moves through a greater distance, or faster, 



174 



AMERICAN MACHINIST GEAR BOOK 



than a point in the pitch circle. Fig. 121 shows a single- threaded hob having 
only one row of cutting edges O, the teeth, or the threads, extending nearly 
around the hob. Let the teeth in the wheel P be shaped as if they had been cut 
to the full depth with the cutting edges of the hob, and in a low-numbered 
wheel we shall have tooth faces shaped as shown. Now, place this gear in mesh 
with the hob at the cutting edges, turn the hob in the direction of the arrow, 
and we shall soon find that the tooth faces of the wheel will interfere with the 
hob threads, as shown at N N, in consequence of the faces N N moving at a 
different speed from the pitch circle. There would also be interference upon 




PIG. 121. A SINGLE-THREADED HOB WITH ONE ROW OF CUTTING EDGE. 



the flanks of the gear, but it was thought that the cut would be quite as clear if 
the showing of this flank interference were not attempted. Only a slice section 
of the wheel is shown at P; in the real wheel we should have a still greater inter- 
ference at the outer part of the teeth T T. In moving along a straight path, 
from R to S, a close-fitting tooth Q would not interfere; but interference would 
begin as soon as Q was moved in a curved path like that of a gear tooth. 

From this consideration of Fig. 1 2 1 we should conclude that it is impractical 
to hob a wheel of few teeth with a hob having only one row of cutting edges, 
like the one shown. Even though we reduce the hob threads back of the cutting 
edges enough to clear the teeth of the wheel, so that it will be possible to hob 
the wheel, we shall not shape the teeth so that they will run correctly with the 
worm. 

Another illustration of interference may be seen in Fig. 122. Let a small gear 
be cut, as shown, with a cutter that is shaped like a gear, as might be done in a 
Fellows gear shaper. Let every rotative movement of the gear, in order to take 



WORM GEARS 



175 



another cut, be through exactly one tooth, a cutter tooth always cutting on the 
line of centers, as shown. In this way cut to the full depth, moving the gear ex- 
actly one tooth at every setting. In our experimental cutting we can let the 
cutter rotate through one tooth at every movement of the gear, or we can let 
the cutter remain stationary, so far as rotation is concerned. When we have 
cut a few spaces to the full depth, we shall find that they are shaped as shown 
in Fig. 122, the spaces below the pitch line merely fitting a cutter tooth upon 

the line of centers without any en- 
veloping or shaping of the gear teeth, 
as there would be in the ordinary 
working of the Fellows gear shaper. 
Now let us stop the cutter, leaving 
its cutting edges just above the side 
of the gear, and try to rotate the gear 





FIG. 



122. ABSENCE OF ENVELOPING 
ACTION IN GEAR CUTTING. 



FIG. 123. A DOUBLE-THREAD HOB WITH 
TWO ROWS OF CUTTING EDGES. 



with the cutter in mesh, just as if they were a pair of gears, and we shall at 
once see that the teeth of the cutter interfere back of the cutting edges, as 
we should suppose from a mere inspection of Fig. 122. 

The same kind of interference that we saw in a single- threaded hob, Fig. 121, 
will occur in a double-threaded hob that has only two rows of cutting edges if 
they are evenly spaced and are parallel to the axis. This can be understood 
from Fig. 123. Any tooth or thread U is exactly opposite another tooth u f 
because the thread is double, one thread starting at the end half way around 
from the other thread. One row of cutting edges U V will pass through the 
spaces cut by the other row u v in the same position as regards the worm-wheel 
teeth, and in consequence the backs of the teeth in both rows will interfere, as 
in Fig. 121. 



176 AMERICAN MACHINIST GEAR BOOK 

A three-threaded hob with three evenly spaced rows of cutting edges will 
interfere, and so on. 

From a careful consideration of the foregoing we arrive at the general prin- 
ciple — The spacing of a hob must not be equal to the circumferential distance oc- 
cupied by either one or to any whole number of threads. 

The more teeth there are in a worm-wheel the more teeth it is possible to 
have in contact with the worm threads at one time, in a worm that is long 
enough, and in consequence a long hob can possibly cut upon enough teeth at a 
time; or, what is the same thing, it can cut every tooth in enough positions so 
that even with only one row of cutting edges it can shape the teeth smooth and 
without interfering. In practice, however, it is never safe to trust to only one 
row. 

My hob has 43 threads and 21 lags or cutting rows. I had spaced the lags 
4 2 3- of the circumference apart, which gave just two thread spaces to each lag. 
Hence, so far as the shape of the worm-wheel teeth is concerned I was not doing 
any different with the 21 lags (shown in Fig. 118) than I could have done with 
only one lag. 

Another body was made for the hob. The lags were spaced evenly, 21 in the 
circumference, which gave 2^ T thread spaces to each lag. This arrangement 
afforded twenty-one positions of lags. To accommodate these positions steps 
were provided, as seen at H. The hob was successful." 

RELIEVING A SPIRAL FLUTED HOB WITHOUT SPECIAL FIXTURES* 

Special fixtures are not necessary to relieve the teeth in a spiral fluted hob. 
This may be accomplished by indexing for a greater number of flutes than are 
actually contained in the hob. 

Let L = lead of hob. 

L x = lead of flute milled in hob. 
C = pitch circumference. 

I = distance gained by spiral flute in one revolution. 
C = circumferential length of each flute. 
N = number of flutes to be added. 







c 




A 


= 




c 


1 




C 




/ 




4 


L 


N 


= 


~c 


• 



*R. J. Briney. 



WORM GEARS 



177 



If N turns out an inconvenient figure it may be changed to the nearest whole 
or fractional number and the lead of flute (L') changed to suit as follows: 

C 



u = 



NC 



jL. 



Example: 

What will be the proper index for the relieving attachment for a hob 4 inches 
pitch diameter and 8-inch lead, number of flute cut in hob 5. 

C = 7T 4 = I2.5664. 



V = -z- c = 



12.5664 



X 12.5664 = 19.739 inches. 



/ = 



L 

c L= 12,5664 x 8 = inches> 

L i 19-739 



v a 



5-088 
12.5664 = 2 



25 



Substituting 2 for 2 , makes our index 5 + 2 = 7, instead of 5. 

Since the value N is changed from 2 _ to 2, we must change the lead of 
flutes to correspond. 2 5 

r 12. 5664 



, T „. L = v , 12.^664 X 8 = 20 inches. 
N C 2 X — - 



REDUCING THE DIAMETER OF 
WORM GEARS 



Increasing the pitch diameter in order 
to avoid undercut is not good practice, 
as it tends to shorten the life of a gear, 
instead of lengthening it. By referring 
to Fig. 124 it is plain that the pitch of 
a worm gear at C is greater than at A 
and, since this pitch the worm can only 
be made to correspond with the pitch 
of the gear at one point, generally A, 
there must necessarily be a great 
amount of friction, with the necessary 
loss in efficiency at B and still more 
at C. 

The efficiency and life of worm gears 
is greatly increased, therefore, by mak- 




fig. 124. 



CORRECTED DIAMETER OP WORM 
GEAR. 



178 



AMERICAN MACHINIST GEAR BOOK 



ing the diameter and, therefore, the pitch of the gear to correspond with the 
pitch of the worm at point B, or the medium pitch diameter of gear. This will 
reduce the pitch diameter the following amount, it being assumed that angle 
of face <£, is 30 degrees; or it can readily be found by a careful layout. 

Corrected pitch diameter of worm gear = D f — 2 (d'—d' 0.97). 

There are in reality as many different pitch diameters between A and C as 
we would care to take sections, as the pitch is changing constantly. For our 
illustration, however, but the three main points have been considered. 



GENERAL MANUFACTURING PROCESSES 

The worm gear is first gashed out by a cutter, approximating the outside 
diameter of the hob for about two-thirds the full depth of the finished tooth, 
or else a taper hob of slightly smaller angle than the worm is used for roughing 
out the worm-wheel teeth. The finishing hob is then placed on the cutter 
spindle and dropped as far as possible into the gashed out tooth. The hob 
then completes the gear, driving it around and finishing the teeth at the same 
time. In starting the finishing out, care must be taken to prevent the teeth 




Angle of 
milling 
machine 



table 



FIG. 125. CUTTING WORM GEAR WITH A 
HOB OF A DIFFERENT ANGLE FROM 
THE ENGAGING WORM. 



of the hob locking on some sharp corner left by the gashing cutter. Gears 
which have been roughly cut with taper hobs avoid this danger to a great 
extent. 

It is always advisable when hobbing the gear wheel teeth to take a hob 
that is similar to the worm to be employed, but when such a hob is not avail- 
able one that is somewhat larger or smaller than the worm can frequently 



WORM GEARS 



179 



be used. See Fig. 125. By offsetting the axis of the hob, as shown in the 
figure, correct teeth can frequently be cut — in fact, it is sometimes possible 
to cut a right-hand wheel with a left-hand hob or vice versa by swinging the 
gear around until the angle of its thread corresponds with the angle of the 
hob. 

STRAIGHT-CUT WORM GEARS 

A modification of the ordinary type of worm and gear has been success- 
fully employed in which the gear teeth are cut in a straight path, like a spur 
gear. See Fig. 126. 




FIG. 126. STRAIGHT-CUT WORM GEAR. 

Advantages are claimed for this construction. It permits side adjust- 
ment which is impossible with the ordinary type of worm gear, and the con- 




FIG. 127. CUTTING STRAIGHT-CUT WORM GEARS ON MILLING 

MACHINE. 



tact is believed to be better as the pitch of the wheel corresponds with that 
of the worm the full width of the face. A disadvantage of the construction 



180 AMERICAN MACHINIST GEAR BOOK 

is that the helix angle of the worm that can be used with such a gear is limited 
to one of about 15 degrees. 

This construction is often used for elevator service as it avoids a certain 
amount of the vibration that is practically unavoidable when employing 
gears of the hobbed type. Spacing errors are easier to avoid when cutting 
the plain straight teeth. 

The teeth of the straight-cut worm gears are cut on milling machines, 
either of the plain or universal type. The table travels at right angles to 
the line of the cutter, the work being set up at the angle of the teeth in the 
worm gear. See Fig. 127. 

Gears of this type should be laid out like spiral gears so that a standard 
spur-gear cutter may be employed to cut the teeth. 

MATERIALS 

The constant sliding between the thread surface of the worm and the face 
of the gear teeth limits and controls the materials or combinations of mate- 
rials of which the worm and gear may be constructed. The worm being the 
active member as far as movement is concerned when slipping past the gear 
teeth, its thread is subject to constant and more destructive abrasion than 
are the teeth of the gear. The gear teeth though not subject to constant 
wear, coming into rubbing contact with the worm but occasionally, must 
nevertheless possess good wear-resisting qualities without being so hard as 
to increase unduly the wear on the worm thread. This relationship 
between the wearing qualities of the worm and of the gear is of the utmost 
importance. 

It is generally recognized that the best materials from which a worm and 
a worm gear can be constructed, in order to realize good wearing qualities, 
high efficiency, etc., are case-hardened steel for the worm and phosphor- 
bronze for the gear. This combination gives excellent results. 

Attempts have been made to substitute manganese-bronze as a material 
from which to make the gear teeth — -usually only the teeth and rim of a 
worm gear are constructed of bronze, the hub, arms, etc., being made of cast 
iron or other cheaper material — -but with disappointing results. The man- 
ganese-bronze was unsuitable on account of its hardness. 

An excellent phosphor-bronze to employ for worm gears consists of: 
Copper, 80 parts; phosphorus, 1 part; tin, 10 parts. 

The steel for the worm may be almost any low-carbon steel, which can 
be readily case-hardened and does not contain more than 3 or 3.5 per cent, of 
nickel nor more than 0.16 to 0.18 per cent, carbon. In case-hardening such 



WORM GEARS 181 

a steel it should be carried to such a point that the scleroscope indicates a 
hardness of from 60 to 70. 

The hardened steel worm should be carefully ground and trued up, but 
such operations are simple and evident for any standard type of worm. 

POWER AND EFFICIENCY OF WORM GEARING* 

In view of the good results now being obtained with worm gearing, the old 
prejudice against that form of gearing, on account of its supposed low efficiency 
and short life, is dying out. These good results are the outcome of the applica- 
tion of principles which are by no means a late discovery, and it is expected 
that what follows will contain much that to some readers is not new. At the 
same time it is an undoubted fact that the best practice with worms is under- 
stood by but few, relatively speaking, and the corroboration of the theory by 
examples from practice which follow is believed to be new. No better illus- 
tration of the fact that good practice with worm gearing is not yet widely under- 
stood could be given than the statement in a recent and excellent work on gear- 
ing that "the diameter of the worm is commonly made equal to four or five 
times the circular pitch," the fact being that such proportions are distinctly 
bad if the worm is to do hard work. 

It should be stated at the beginning that while what follows is not offered as 
a presentation of all the data necessary for assured success with worms under 
all conditions, it is hoped to make the general conditions of successful practice 
plain, and to present the "state of the art" as it exists to-day. 

The essential change in practice which has improved the results obtained 
with worm gearing has been an increase in the pitch angle over what was former- 
ly considered proper. There is no doubt whatever that this change has in- 
creased the efficiency of the gear, and, what is of more importance, has reduced 
the tendency to heat and wear. This is not only a fact, but it is a sound 
conclusion from theoretical considerations, which might have been predicted 
under proper examination. 

THEORY OF WORM EFFICIENCY 

The reason why an increase of pitch, other things being equal, or, in other 
words, an increase of the angle of the thread, gives these results, will be under- 
stood from Fig. 128. If a b be the axis of the worm and c d a line representing 
a thread, against which a tooth of the wheel bears, it will be seen that if the 
tooth bears upon the thread by a pressure P, that pressure may be resolved into 
two components, one of which, e f, is perpendicular, while the other, e g, is 
* F. A. Halsey, in the American Machinist. 



l82 



AMERICAN MACHINIST GEAR BOOR 




parallel to the thread surface. The perpendicular component produces friction 
between the tooth and the thread. The useful work done during a revolution 
of the thread is the product of the load P and the lead of the worm, while the 
work lost in friction is the product of the perpendicular pressure e f, the co- 
efficient of friction and the distance 
traversed in a revolution, which is the 
length of one turn of the thread. Now, 
if the angle of the thread be doubled, 
as indicated, the load P remaining the 
same, the new perpendicular component 
f hoi P will be slightly reduced from the 
old value ef, while the length of a turn 
of the thread will be slightly increased. 
Consequently their product and the lost 
work of friction per revolution will not 
be much changed. The useful work per 
revolution will, however, be doubled, 
because, the pitch being doubled, the 
distance traveled by P in one revolution 
will be doubled. For a given amount 
of useful work the amount of work lost 
is therefore reduced by the increase in 
the thread angle, and, since the tendency to heat and wear is the immediate 
result of the lost work, it follows that that tendency is reduced. For small 
angles of thread the change is very rapid, and continues, though in diminishing 
degree, until the angle reaches a value not far from 45 degrees, when the con- 
ditions change and the lost work increases faster than the useful work, an in- 
crease of the angle of the thread beyond that point reducing the efficiency. 

This general consideration of the subject shows the principles at the bottom 
of successful worm design, but a more exact examination is desirable. Accord- 
ing to Professor Barr the efficiency of a worm gear, the friction of the step being 
neglected, is: 

tan a ( 1 — / tan a) 

g = ■ = — 

tan a + / 
in which 

e = efficiency, 

a = angle of thread, being the angle df i of Fig. 128, 
/ = coefficient of friction. 

To study the effect of the step, a convenient assumption is that the mean 
friction radius of the step is equal to that of the worm. This assumption 



FIG. 121 



THE PRINCIPLE OP WORM 
EFFICIENCY. 



WORM GEARS 1 83 

would be realized only in cases where the step is a collar bearing outside the 
worm shaft, and the preceding and following formulas therefore represent 
extreme cases, one of a frictionless step, which would be approximated by a 
ball bearing, and the other of a step having about the extreme friction to be 
met with. Most actual cases would therefore fall between the two. Again, 
according to Professor Barr, the efficiency of a worm and step on the above 
assumption is: * 

tan a (1 — / tan a) . 

e = r — 1 (approximately). 

tan a + 2] 

Notation as before. 

These formulas give no clear indication of the manner in which the efficiency 
varies with the angle, and Chart 10 has been constructed to show this to the 
eye. The scale at the bottom gives the angles of the thread from o to 90 
degrees, while the vertical scale gives the calculated efficiencies, the values 
of which have been obtained from the equations and plotted on the diagram. 
The upper curve is from the first equation, and gives the efficiencies of the 
worm thread only; while the lower curve, from the second equation, gives the 
combined efficiency of the worm and step. In the calculations for the dia- 
gram it is necessary to assume a value for/, and this has been taken at 0.05, 
which is probably a fair mean value. The experiments made by Mr. Wilfred 
Lewis for Wm. Sellers & Co. showed an increase of efficiency with the speed. 
The present diagram may be considered as confined to a single speed, and 
at the same time is not to be understood as showing the exact efficiency to be 
expected from worms, but rather to exhibit to the eye the general law con- 
necting the angle of the thread with the efficiency. 

The curves will be seen to rise to a maximum and then to drop. The exact 
values of the angle of thread to give maximum efficiency may be easily found 
by the methods of the calculus, the results being: 

For worm thread alone the efficiency is at a maximum when 

tan a = V 1 + / 2 - /. 
Substituting the value of/ (0.05) used in calculating the diagram, this becomes 
tan a for maximum efficiency = 0.9512, and by referring to a table of 
natural tangents we find that a for maximum efficiency = 43 ° 34. 
Similarly for the worm and step the result is tan a for maximum efficiency = 

\/ 2 +/ 4 2 — 2/, which for/ = 0.05 = 1. 318, 
and a table of tangents tells us again that a for maximum efficiency = 52 2 49 r . 
Of more importance than the angle of maximum efficiency is the general 

*In Professor Barr's formulas it is assumed that the worm thread is square in section. 
Thread profiles in common use affect the results but little. 



1 84 



AMERICAN MACHINIST GEAR BOOK 



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Efficiency Percent 



WORM GEARS 185 

character of the curves, of which the most pronounced peculiarity is the extreme 
flatness, showing that for a wide range of angles the efficiency varies but little. 
Thus, for the upper curve there is scarcely any choice between 30 and 60 
degrees of angle, and but little drop at 20 degrees. 

At first sight the lower curve might be thought the more useful of the two, 
as it includes the effect of the step, but a little consideration will show that 
this is not the case. For most cases in which worms are used the efficiency 
of the transmission, as such, is of very little account. What the designer 
concerns himself with is the question of durability and satisfactory working, 
and the results to be expected in this respect are best shown by the upper 
curve, in which high efficiency means a durable worm. Throughout this 
discussion, in fact, the chief significance of efficiency lies in the fact that low 
efficiency means rapid wear, and vice versa. 

EXPERIMENTAL CORROBORATION OF THE THEORY 

The experiments of Wm. Sellers & Co., before referred to, go far to confirm the 
soundness of the above views. From the present standpoint it is unfortunate 
that those experiments did not cover a wider range of worm-thread angles — 
those actually used being 5 degrees, 7 degrees, and 10 degrees. Other experi- 
ments were, however, made on spiral pinions of higher angles, spiral pinions 
being understood by Mr. Lewis to mean those pinions having the mating gear 
a true spur, the pinion shaft being at a suitable angle with the gear shaft to 
bring the pinion in proper mesh — a construction which is exemplified in the 
well-known Sellers planer drive. Mr. Lewis gives a formula by which the 
efficiencies of worms can be calculated from those for spiral pinions, and in the 
absence of direct experiments on worms of high angles, his results for spiral 
pinions have been modified by this formula to read for worms. The results 
for the two forms of gearing differ by less than five per cent, for the extreme 
case of his experiments. To compare the results obtained by Mr. Lewis with 
Professor Barr's formula, a speed has been selected from the experiments 
giving the nearest coefficient of friction to that used in obtaining the curves of 
Chart 11. The results have been plotted in Chart 11, where they appear as 
small crosses, and will be seen to have a very satisfactory agreement with the 
lower curve, with which they should be compared, as the steps of the worms 
used by Mr. Lewis were of the usual pattern without balls. 

The variation of the coefficient of friction with the speed lends an interest 
to Chart 11, which is a series of curves obtained from the results published by 
Mr. Lewis in the same manner as the crosses of Chart 10, the curve for 20 feet 
velocity being in fact the same as that appearing as crosses on Chart 11. The 
other curves of Chart 11 are obtained from those of Mr. Lewis, and cover a 



i86 



AMERICAN MACHINIST GEAR BOOK 



range of velocities from 3 to 200 feet per minute at the pitch line, as noted at 
the right. In this diagram the results obtained by Mr. Lewis on worms are 
plotted direct, but the experiments on spiral pinions have been modified as 
explained above. Inspection of the curves shows that while there is a pro- 
gressive increase of efficiency with the speed, there is, nevertheless, not much 



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Angle of Thread- Degrees 



*B 



to 8 



CHART II. 



RELATION BETWEEN THREAD ANGLE, SPEED AND EFFICIENCY WITH CASES 
FROM PRACTICE. 



probability, or indeed room, for further improvement beyond the speed of 
200 feet per minute. It will furthermore be seen that the efficiency drops off 
much less for low angles of thread at high speeds than at low. 

In interpreting this diagram, it should be remembered that the durability of 
a worm depends upon the amount of power lost in wear, and not upon the 
percentage so lost. The ability of a given worm to absorb and carry off the 
heat due to friction is fixed, and does not vary with the speed. That is, a given 
worm running at 100 revolutions under a given pressure can carry off as much 
friction heat as the same worm at 200 revolutions, while it, under the same 
pressure, would transmit but one-half the power in the former case that it 
would in the latter. In other words, the percentage of lost work might be 
twice as much at the lower speed as at the higher without increasing the tend- 
ency to heat. 



WORM GEARS 187 

The increase of efficiency with the speed is a valuable property of worms, 
and enables them to do much more work than they otherwise would. Thus 
the 20 degree worm at 20 feet per minute lost 21 3^ per cent, of the work in 
friction. Increasing the speed to 40 feet doubled the work applied, and, 
had the efficiency remained constant, would have doubled the friction heat to 
be dissipated. In point of fact, this increase of speed diminished the per- 
centage of loss to 17, and the amount of loss and heat, instead of being doubled, 
was only increased in the ratio of 160 to 100. It is plain from the diagram, 
however, that this action does not continue much beyond a velocity of 200 
feet per minute, beyond which the amount of loss must be more nearly pro- 
portional to the speed, and this doubtless has some connection with the fact 
observed by Mr. Lewis that 300 feet per minute is the limit of speed when the 
gears are loaded to their working strength, and that the best conditions are 
obtained at about 200 feet per minute. It is proper to add, however, that in 
the cases from practice given later there are three which have been made 
repeatedly, and which are conspicuously successful, in which the velocity 
exceeds 600 feet, and one in which it exceeds 800 feet. No doubt, in all such 
cases, if the pressure on the teeth could be known it would be found to be light. 

It will be seen that an increase of speed for any worm under constant pres- 
sure leads to an increase of friction work, and the limit is reached when the 
worm is no longer able to carry off the heat generated fast enough to prevent 
undue rise in temperature. Furthermore, this limiting speed depends upon 
the pressure, it being higher for low pressures than for high. A worm having 
an angle which might be successful at low speed may fail at high speed; but it 
would seem that any worm which is successful at high speed should also be 
successful at low, which is in accordance with mechanical instinct. 

There are, it will be observed, two methods of increasing the pitch angle. 
The diameter may be kept constant and the pitch be increased, or the pitch may 
be kept constant and the diameter be reduced. From a mathematical stand- 
point, these two methods are identical; that is, at a given pitch-line velocity 
a worm of a given angle should have the same efficiency, regardless of the diam- 
eter; but in a mechanical sense the methods are not identical. The worm of 
the larger diameter would naturally have a gear of wider face, and the pair, 
having greater area of tooth surface in contact, would carry a larger load. 

EXAMPLES FROM PRACTICE 

It is impossible to say who was the first to recognize the significance of the 
pitch angle as a factor in the satisfactory performance of worm gearing, but it 
may be mentioned as a matter of interest that the exhibit of the Hewes & 
Phillips Iron Works at the Newark Industrial Exhibition of 1873 included 



1 88 AMERICAN MACHINIST GEAR BOOK 

several worm-driven planers, in which the worms were double- threaded and 
had a pitch angle of 15 15', a pitch diameter of 33^2 inches, a lead of 3 inches, 
and a speed cutting of 256 and backing of 640 r. p. m., which give pitch-line 
velocities of 237 and 590 feet. This worm was successful, and was many 
times repeated; but later on Hewes & Phillips were struck by the high-belt 
speed idea, and in order to increase the belt speed they changed the worm to 
6.16 p. d., iM inches pitch, single thread; speed cutting, 446, and backing 
1,110 r. p. m., giving a pitch angle of 5 15' and pitch-line velocities of 720 
and 1,780 feet. This worm was a failure, and was soon changed to 6.16 p. d., 
3^ inches lead, double-thread, speed cutting, 281, and backing 700 r. p. m., giv- 
ing an angle of io° 15' and pitch-line velocities of 452 and 1,130 feet. This worm 
did better than the last, but not so well as the first. By this time the lesson 
was learned, and Hewes & Phillips set out to use a worm of 30 degrees pitch 
angle. Structural considerations, however, prevented the use of so high an 
angle and they compromised on 20 degrees, the final worm resulting from this 
experience having a pitch diameter of 2.63 inches, with 3 inches lead, quad- 
ruple thread, the speed cutting being 300 and backing 700 r. p. m., giving 
pitch-line velocities of 205 and 480 feet, and this remained the standard angle 
as long as these planers were manufactured. 

The writer has seen one of these 20-degree worm gears, opened up after 
twelve years' use, and the wear disclosed was very slight — oiq shoulder being 
in existence. As a result of the experience outlined above, this house adopted 
the standard practice of the worms as small as possible in diameter, and 
giving the threads in all cases a pitch angle of 20 degrees. The form of tooth 
used was the epicycloidal, while the materials used were hard cast iron for the 
gear and case-hardened open-hearth steel for the worms. 

These Hewes & Phillips worms are plotted in Chart n as crosses 1, 2, 3, 4, of 
which 1 is the 15 15', 2 the 5 15', 3 the io° 15', and 4 the 20 , the first and last 
being successes, and the second and third failures. 

In plotting these worms, and all others having pitch-line velocities above 200 
feet, the crosses are placed near and above the 200 feet curve. It is unfortunate 
that we have no curves for higher speeds, but Mr. Lewis recommends the use 
of the 200 feet line for all higher speeds. Leaders connecting different crosses 
indicate the same worm at different speeds in all cases. The letters s and j 
on the diagram mean success or failure in all cases. 

Fig. 129 is a drawing of a worm 3 (failure) and Fig. 130 shows worm 4 (suc- 
cess), and no more instructive pair of drawings could be imagined than these. 
The pitches are not far different, and what difference there is is in favor of the 
larger worm. The duty is the same, the gears are of about the same diameter, 
and the revolutions per minute are nearly the same. The essential change is 



WORM GEARS 



189 



in the increase of the pitch angle by a reduction of the diameter, and this 
changed failure to success. 

The Newton Machine Tool Works use worm gearing in many of their 
machines, notably their cold saw cutting-ofT machines. In the earlier machines 
of this class the worm had a pitch diameter of 2^ inches, with a pitch of 1 
inch, single thread, the revolutions per minute being 765. These figures give 




FIG. 129. HEWES & PHILLIPS 
UNSUCCESSFUL WORM. 



PIG. T30. HEWES & PHILLIPS 
SUCCESSFUL WORM. 



a pitch angle of 6° 20', and a pitch-line velocity of 572 feet. This machine 
could be operated, but not with satisfaction on account of the heating and 
short life of the worm. The worm was then increased in lead by making it 
double- threaded, giving a pitch angle of 12 30', the speed being reduced to 
500 revolutions per minute, giving a pitch-line velocity of 375 feet. The 
change proved to be a great improvement, heavier work than was before 
possible being done after the change without distress or difficulty, and this 
worm has since been applied to a large number of machines with entire success. 
A still later worm used on these machines has a pitch diameter of 3!^ inches 
and a lead of 4 inches, triple threads, giving a pitch angle of 18 15', and this 
is found to be a still further improvement. This last worm is used on a wide 
variety of machines and at a variety of speeds from 40 to 680 r. p. m., giving 
pitch-line velocities of from 40 to 685 feet, and with uniformly good results. 
In many cases it is used without an oil cellar, though for comparatively light 
work. The form of thread used is the involute, and the material is hardened 
steel for the worm and bronze for the wheel. These Newton worms appear in 
Chart 11 as 5, 6, 7, of which 5 is nearly, a failure, while 6 and 7 are entirely 



190 



AMERICAN MACHINIST GEAR BOOK 



successful. The second Newton worm — the one appearing in Chart 11 as 6 — 
is shown in Fig. 131. 

Another habitual user of worms is John Bertram & Sons, of Dundas, Ontario, 
Canada, who employ them in all their planers, and use largely a worm of 3.18 
inches pitch diameter, 4 inches lead, quadruple threads, the speed cutting 
being 186 and reversing 744 r. p. m. These figures give a pitch angle of 22 
degrees, and pitch-line velocities of 155 and 620 feet. This worm appears in 
Chart n as 8, the vertical position for the higher speed being again uncertain. 
These worms are highly successful, as the writer knows from repeated obser- 




r-^n 



FIG. 131. THE NEWTON WORM AND STEP. 



FIG. I32. THE BERTRAM WORM. 



vation. Both worm and wheel are of cast iron, the thread being Brown & 
Sharpe standard. The Bertram worm is shown in Fig. 132. In reading this 
drawing it should be remembered that the conventional representation of a 
worm, with the threads shown by straight lines, shows a larger apparent pitch 
angle than the true one, as shown by a true projection. 

Another case of failure was a worm drive applied to a large boring machine, 
the worm being 12 inches pitch diameter, 8 inches lead, quadruple thread, 
speed 80 r. p. m. and above, worm of forged steel, wheel of bronze, oil cellar 
lubrication. These figures give a pitch angle of 12 degrees and a pitch-line 
velocity of 250 feet. This worm is located on Chart 11 as 9. 

Still other cases of change from failure to success are supplied by Mr. Jas. 
Christie, of the Pencoyd Iron Works. The first of these relates to a boring 
machine, which was, by the makers, supplied with a worm drive having a worm 
of 5^ inches pitch diameter, i}4 inches pitch, single thread, steel worm and 
cast-iron wheel, average speed 150 r. p. m. These figures give a pitch angle 
of 5 degrees and a pitch-line velocity of 215 feet. This was a failure, but was 
successfully replaced by a worm of 4H inches pitch diameter, 2 Y A inches lead, 
and the same number of revolutions, which figures give a pitch angle of 9 



WORM GEARS 191 

15' and a pitch-line speed of 190 feet. These two worms appear as 10 and 11. 
This successful worm lies in the region of unsuccessful ones, but the influence 
of the increased lead angle is unmistakable. The fact of its success is probably- 
due to the pressure on the teeth being well below the working strength, or to 
the speed being moderate, or both. 

The second case, of which the data were supplied by Mr. Christie, relates 
to two heavy milling machines, in which the cutter spindles were driven by 
worms 6 inches pitch diameter by i3^ inches pitch, single thread. It was 
found that the cutters could be run much faster than was orginally contem- 
plated, and the worms were consequently speeded up to about 500 r. p. m. 
In these machines cast-iron worm-wheels were speedily destroyed, while 
hardened steel worms and bronze wheels would last about a year. Later 
two more machines were built having steel worms and bronze wheels, the 
Worms being 4^ inches pitch diameter by 5 inches lead, quadruple threads, 
speed 280 r. p. m. These worms have been in use six years, and are described 
as being "good as new." The data given for the first worm give a pitch of 4 
30' and a pitch-line velocity of 785 feet. It appears in Chart 11 as 12. 
The pitch angle of the second worm is 19 30', and its pitch-line velocity 328 
feet. It appears in Chart 11 as 13. 

Mr. Christie has made many successful changes, of which these are typical, 
and he now uses worms with great freedom and success. His general con- 
clusion is that good worms begin with those having the pitch about equal to 
the diameter, giving a pitch angle of 17 15'. 

Another equally striking case of success accompanying an increase of the 
pitch angle is supplied by Mr. W. P. Hunt, of Moline, 111., who says: 

"In building a special double-spindle lathe I wished to use a worm drive, 
and having a single thread M-inch pitch hob, 2 % -inch outside diameter, I 
decided to work to that, and made my gear with 26 teeth, giving a 
speed reduction of 26 to 1. The worm was to run at 460 revolutions per 
minute, but upon starting the machine I found it impossible to keep the worm 
and gear cool, and the belts would not pull the cut. 

"Accordingly I decided to make a new worm and hob having the same out- 
side diameter as the one first tried, but with double thread and i-inch pitch, 
2-inch lead and a new gear having 48 teeth, giving me a speed reduction of 24 
to 1, or less than at first. 

"Upon starting the machine with the new worm and gear, not only did it 
run perfectly cool, but the belts have ample power. We use graphite and oil 
on the worm, and it is not enclosed." 

Mr. Hunt does not give the pitch diameter of his worms, but assuming the 
threads to have been in accordance with the Acme standard, the pitch diameters 



192 



AMERICAN MACHINIST GEAR BOOK 



are 2%, and 2M inches respectively, the thread angles being 5 44' and 15 48', 
and the pitch-line speeds 286 and 271 feet per minute. Mr. Hunt's worms 
are plotted in Chart 11 as 17 and 18. 

Three other cases of successful worms under heavy duty are found in milling 
machines which have been repeated many times. The first two worms would 
ordinarily be described as spiral gears. The shafts are at right angles. 

The first of these, which appears as 14 in the diagram, has a pitch diameter 
of 2M inches, a pitch angle of 45 °, and a speed varying between 180 and 945 
r. p. m., giving pitch-line velocities of 106 to 555 feet per minute. Both gears 
are of cast iron. The second, 15 in the diagram, is of the same style, and has 
the same pitch diameter, with speeds varying between 90 and 472 r. p. m., 
giving pitch-line velocities of 53 to 277 feet per minute. The third, 16 in the 
diagram, is a true worm, 2M inches pitch diameter, lead 1.333, triple thread, 
speed 200 to 1,442 r. p. m., bronze wheel and hardened steel worm. These 
figures give a pitch angle of io° 45', and a pitch-line velocity of 118 to 845 
feet per minute. While this worm is entirely successful, it was at first a failure. 

LIMITING SPEEDS AND PRESSURES 

A very important point connected with worm design, and one on which data 
are very scarce, is the limiting pressures for various speeds at which cutting 
begins. The paper by Mr. Lewis contains some information on this subject, 
and the accompanying table supplied by Mr. Christie, from experiments made 
by him, supplies the most definite additional data on the subject known to the 
writer. In all cases the worms were of hardened steel and the worm-wheels 
of cast iron. Lubrication by an oil bath. 



Revolutions per minute .... 

Velocity at pitch line in feet 
per minute 



Limiting pressure in pounds 



SINGLE-THREAD WORM 

l" PITCH 

2^8 PITCH DIAMETER 



128 


201 


272 


96 


150 


205 


1,700 


1,300 


1,100 



425 

320 

700 



DOUBLE-THREAD 

WORM 2" LEAD 

2j^ PITCH 

DIAMETER 



128 


201 


96 


150 


,100 


1,100 



272 

205 

1,100 



DOUBLE-THREAD 

WORM 23^" LEAD 

4^ PITCH 

DIAMETER 



201 


272 


23s 


319 


,100 


700 



425 

498 

400 



Limiting Speeds and Pressures op Worm Gearing. 

There is real need of a comprehensive series of experiments on this subject. 
It is obvious enough that a worm, otherwise well designed, might fail from hav- 
ing too high a speed for its load. Were such data at hand it would seem that 



WORM GEARS 193 

with existing knowledge of the influence of the angle of the thread, worm design 
might be made a matter of comparative certainty. Especially should the be- 
havior of worms at speeds above 200 feet per minute be subjected to further 
experiment, as it is frequently necessary to use speeds above that figure, and 
there can be no doubt that higher speeds are entirely feasible if suitable pres- 
sures accompany them. The speed as a factor should be kept in mind equally 
with the pitch angle. A worm may fail because of too high a pitch-line velocity 
as well as because of too low a pitch angle. 

The number of cases cited is too few for certainty in drawing general con- 
clusions, but the testimony is unmistakable in its confirmation of the theory 
of the influence of the angle of the thread. It will be seen that every case having 
an angle above 12 30' was successful, and every case below 9 unsuccessful, the 
overlapping of the successful and unsuccessful worms in the intervening region 
being what is to be expected in the border region between good and bad prac- 
tice. This band of uncertain results is in fact narrower than we would have any 
right to expect from a collection of data from miscellaneous sources, and could 
the inquiry be widened in scope the width of this band would doubtless be in- 
creased. As throwing light on these cases, it should be remembered that case 
16 is known to have been made successful only by careful attention to the 
material used, the first worms made having been failures, and that 3, which is 
near 16 and was a failure, had an excessive speed, while n, at a lower angle 
and a success, had a very moderate speed. At a higher speed 1 1 would prob- 
ably have failed, and at a lower speed 3 would probably have been a success. 
It is believed that Chart 1 1 points out clearly the nature of the worm problem 
and the conditions of success in its solution. 

EFFICIENCY AND TEMPERATURE 

The unavoidable sliding action between the threads of the worm and the 
teeth of the gear represents lost energy which must be measured by the 
amount of friction heat generated. The elevation in the temperature of the 
lubricating oil about a worm drive should then approximate the efficiency of 
the construction. 

The relationship between the efficiency of a journal bearing and its rise 
in temperature is similar and is generally recognized as a reliable indication 
of the efficiency of the bearing. Unfortunately, a number of experiments 
that have been conducted with a view of discovering such relationship in 
connection with worm drives have been rather misleading. This has been 
partly due to inadequate lubrication, but more probably to the fact that both 
liquid and solid friction have been present. 



194 AMERICAN MACHINIST GEAR BOOK 

Comprehensive experiments to ascertain the relationship between effi- 
ciency and temperature of worm drives at the engineering laboratory of the 
Royal Technical High School, Stuttgart, Germany, however, have conclu- 
sively shown that some relationship does exist. The conclusions arrived at 
by the experimenters may be summarized as follows: 

i. The difference in oil temperature is approximately proportionate to 
the tooth pressure when the speed of the worm remains constant. 

2. The tooth pressure decreases according to a definite law with any 
increase in worm speed when the temperature remains constant. (A 
curve depicting such decrease in tooth pressure is of distinctly hyperbolic 
character.) 

As confirmation of the above deductions, the conclusions arrived at by the 
firm of Henry Wallwork & Co., Ltd., Manchester, England, which has con- 
ducted many individual tests to ascertain the laws governing worm-gear 
efficiency, are of particular interest. A. V. Wallwork summarized these 
conclusions, in a letter that appeared in American Machinist, Dec. 5, 191 2, 
as follows: 

"That the efficiency of a correctly designed gear increases under light 
load and decreases under heavy load, and that the temperature rise of the 
oil gives an exact measure of the power loss in the gear." 

EFFICIENCY OF LANCHESTER WORM GEAR 

The Lanchester worm, which is very similar to the Hindley worm, being 
of the hour-glass variety, has been subject to exceedingly exhaustive experi- 
ments at the National Physical Laboratory of Teddington, England, which 
were briefly described in American Machinist, June 12, 1913. 

The engineers making these tests arrived at the conclusions that the 
efficiency of the worm gear itself depends entirely on the condition of the oil 
film between the worm and the wheel and that the efficiency would remain 
practically constant as long as this oil film is perfect. The fall in efficiency 
of the gear at slow speed, which was quite apparent in all tests, was believed 
to be due in part to the reduction in the quantity of oil carried around by the 
worm. 

In conclusion, the report submitted to the Daimler Co. of England, for 
which the tests were made, says: 

"The conclusion to be drawn from these tests is that the efficiency of the 
gears lies between 93 and 97 per cent, under all circumstances, and, taking 
the normal running speed of the worm at 1,000 r. p. m., the efficiency of the 
gears lies between 95 and 97 per cent., only falling slightly below the lower 
figure when the temperature approaches 100 degrees Centigrade." 



WORM GEARS 1 95 

AUTOMOBILE WORM DRIVES 

The prejudice that appears to exist in this country against the worm drive 
for automobiles is not so pronounced in Europe. There does not seem to be 
any reasonable ground upon which to base the objections made by many 
manufacturers in regard to worm-gear transmissions. English manufac- 
turers in particular have secured very gratifying success with worm drives. 
David Brown & Sons, Ltd., Lockwood, Huddersfield, England, claim excellent 
results for their worm-gear auto drives — stating that 95 per cent, efficiency 
is easily realized. E. G. Wrigley & Co., Ltd., Birmingham, England, claim 
for their worm drives an efficiency as high as that of a bevel-gear drive at 
normal speed, equal or greater efficient life and practical silence in operation. 

Ralph H. Rosenberg presented a paper before the Society of Automobile 
Engineers, excerpts of which were printed in American Machinist, Feb. 29, 
191 2, in which he summarized his conclusions on the worm drive for heavy 
power vehicles in part as follows: 

"My contention is — and I have proved it empirically — that, first, the 
tooth angle and angle of lead or advance must coincide, and second, that the 
advance angle fixes the diameter of the worm. It is determined by extend- 
ing the lines describing the flanks of the teeth to points where the intervening 
distance is equal to the lineal pitch times the number of leads; the distance 
from these points to the pitch circle of the gear is the diameter of the worm. 

"I have adopted the following method for determining the width of the 
gear and face of the teeth. They are described by diverging lines from the 
center of the worm, including an angle of 120 degrees. Gears made accord- 
ing to this formula will permit a reasonable amount of variation between 
the pitch circle of the worm and the pitch circle of the gear, the surfaces 
remaining complementary. This is not permissible with any other form and 
allows a certain latitude in manufacture. It is absolutely necessary, however, 
to maintain proper relations of the axes. 

COST 

"I have heard it asserted by those conceding the utility and desirability 
of the worm gear that it was an expensive device. My endeavors to ascertain 
upon what ground this assumption was made and what particular item en- 
tered into the consideration of cost were usually met by general statements. 
So I conclude that the cost-of-manufacture information while not as vague 
as that relative to designing is, nevertheless, indefinite. I take the liberty 
of quoting from E. E. Whitney's paper of June, 191 1, relative to the cost of 
worm-gear construction, wherein he states the worm-gear drive is not a cheap 



196 AMERICAN MACHINIST GEAR BOOK 

device and that the indicated efficiency and durability results cannot be 
expected unless the gears are properly designed, constructed of the best 
materials and adequately mounted in high-grade anti-friction bearings. 

"I concede this statement to cover the essential facts generally, but on 
the question of cost take issue, believing the worm gear to be the cheapest 
form of final drive. It is admittedly true that proper design is essential to 
the success of any mechanism, but it does not follow that proper design will 
entail any expense over and above improper design, so far as it relates to 
the cost of manufacture. In my experience I have found that the materials 
used in the worm and gear are not more expensive than those employed in 
the bevel-gear drive or the side-chain drive, where double reductions are 
used. Furthermore, a distinction should be made between experimental 
work and work of actual production where the facilities are provided for 
executing large quantities. In substantiation of my statement I give the 
following data, taken from records covering the cost of producing a worm 
and gear for a 5-ton truck: 

Bronze ring gear blank $18 . 00 

Steel for worm 9 . 00 

Time to machine worm 2 hours 

Labor on worm, rough- turning 2 hours 

Labor on worm, moulding 2 hours 

Labor on worm, grinding 3 hours 

Ring gear, turning 3 hours 

Ring gear, cutting . 1 hour and 

10 minutes 

"The parts are then in condition for assembling. Concerning the desir- 
ability of the worm gear, I am in accord with Mr. Whitney, and can say that 
I have inspected gears after they have run 120,000 miles and found them in 
excellent condition. Granting that the expense of production is higher, it 
is offset by the greater life of the gear." ' 

There is no question that properly proportioned and constructed worm- 
gear drives can be made with high guaranteed efficiency ; that they are com- 
pact and may be constructed for considerably higher ratios without requiring 
undue space for their accommodation; that their wearing qualities can be as 
great as those of a bevel-gear drive; and that they possess other advantages 
of silence, smooth operation, etc. These numerous advantages, which are 
now lessened by no serious drawbacks, should appeal strongly to the automo- 
bile manufacturer, particularly in the construction of auto trucks requiring 
large reduction in speed between the motor and the driving wheels. 



WORM GEARS 



THE HINDLEY WORM GEAR 



197 



An interesting modification of the standard worm gear is frequently 
encountered in elevator service, and is known as the Hindley worm gear or, 
from its form, the globoid gear. The worm differs from the ordinary straight 
worm because, instead of being cylindrical in shape, it is formed somewhat 
like an hour-glass or spool. The pitch diameter of the worm varies so as to 
coincide with that of the gear for the full length of the worm. The thread 
of the worm is in mesh with several gear teeth at the same time, the worm 
enveloping a section of the gear wheel. See Fig. 144. There is no bottom 




FIG. 144. THE HINDLEY WORM GEAR. 



clearance in this gear and the length of tooth and depth of thread are some- 
what greater than in the common gear and worm, thus increasing contact 
surface and therefore decreasing wear. 

The smallest diameter threads, at the center of the worm, evidently 
engage the gear teeth as do the threads in an ordinary worm gear, the con- 
tact of which is on the pitch line. The threads toward the ends of the worm 
are of greater diameter, however, and as they do not revolve on axes tangent 
to the pitch diameter of the gear, but about the axis of the worm, contact 
is not simply on the pitch line, but on the entire surface of the engaging gear 
teeth. That is, the center teeth are in line contact, while the end teeth 
are subject to surface contact. 

An article by John L. Wood, which appeared in American Machinist, 
June 23, 1 914, takes up in detail the method of calculating the strength of 



198 AMERICAN MACHINIST GEAR BOOK 

Hindley worm gears and the methods of calculating them. Excerpts of 
this discussion follow: 

METHOD OF CALCULATING STRENGTH 

"The following example is given to show a method of calculating the 
strength of this construction: 

"In a carriage the theoretical load to be supported by means of two worms 
in the direction of their axes was 25,000 pounds. 

"For the construction the following dimensions for the worm and rack 
were used: 

WORM DIMENSIONS 

Smallest diameter at root of teeth 1 . 072 inches. 

Corresponding pitch diameter 1 . 348 inches. 

Corresponding diameter at top of thread 1 . 587 inches. 

Obliquity of teeth 15 degrees. 

Number of teeth in contact with rack 7 

Pitch = 0.375 inch- 
Lead = 0.75 inch. 

RACK DIMENSIONS 

Thickness = length of tooth = face of tooth 0.95 inch. 

Pitch diameter 12 . 652 inches. 

Number of teeth in complete circle 106 

Load for each worm and rack 12,500 pounds. 

"Assuming that all of the seven teeth are effective and that each takes 
its proportionate load, we have, 

Load for each tooth — W = 1,785 pounds. 

"Using the Lewis formula for the rack and assuming, as stated before, 

that / = face of tooth = thickness of the rack at the root of the rack teeth, 

we get, 

W = Spfy, 

or 

W 1785 
S = —r = — — = 42,4=50 pounds per square inch. 

Ply 0.375 x 0.95 x o.n8 

"The rack was made of steel with an elastic limit of approximately 53,000 
pounds per square inch, and no trouble was experienced with this rack and 
no change was ever necessary. 

"The worm was made of bronze with a tensile strength of 52,000 pounds 
per square inch. Since its smallest diameter was larger than the face of 



WORM GEARS 199 

the rack, it might have been supposed that the teeth would have been 
sufficiently strong. This proved not to be the case, and the material was 
changed to steel with an elastic limit of 53,000 pounds per square inch, and 
no more difficulties were experienced. 

"In many other places where this Hindley worm is used it has been found 
that the rack can be calculated by using the Lewis formula and that a safe 
rule for calculating the strength of the worm teeth is: 

"Divide the total load to be supported by the number of teeth in contact. 
Assume this load to be applied at the top of the tooth. Consider the tooth 
a cantilever, of which the base is a line drawn tangent to the root of the 
tooth of the smallest diameter of the worm, the length of this base being 
the distance between the points of intersection of this line and the pitch 
line of the tooth, and the width being the thickness of the tooth at the root. 

" Since this method of calculating the worm is not convenient, the following 
rules which have shown satisfactory results are given: 

"Instead of considering as the base of the tooth the chord of the pitch 
diameter, which touches the root of the tooth, assume this base to be twice 
the chord subtended by half of this angle. 

" Call this distance /. Then / = 2 y/Dm, in which D is the pitch 
diameter of the worm, and m is the depth of space below pitch line. Let 
p = circular pitch of worm and rack, then m = 0.3683 p. And 

/ = 2 y/D X 0.3683 p = 1.2 y/pD. 

"Now substitute for/ in Lewis formula, and we have for the worm W = 
1.2 Spy y/pD, and for the rack W = Spfy, in which S, p, y have values the 
same as for any other gears, while / is the width of the rack, and D the pitch 
diameter of the worm; y is the same for the worm as for the rack. 

"It is evident that a gear of this kind must be made most accurate. This 
is especially so since the teeth in the rack are cut with a hob of as many teeth 
as there are in the worm itself. If any inequalities are found in the shape of 
the teeth in the hob, it will be seen that the rack teeth being cut to suit the 
largest section of the hob tooth and the worm being exactly like the hob, all 
the load might be taken on one tooth only. 

"The following method is the manner of manufacturing these gears at the 
Rock Island Arsenal: 

OPERATIONS FOR MAKING WORM 

"First: Cut off stock y$ inch longer than the dimension required for the 
length of the worm and then center the part. Second: Between centers of 
engine lathe rough-turn all diameters }^ inch large and all length dimensions 



200 



AMERICAN MACHINIST GEAR BOOK 



34 6 mcn l° n g- Note: There should be i inch of stock left on the length of 
threaded part to be cut off after thread has been cut. This is to insure against 
any error created by the spring of thread tool on entering and leaving the 
cut and to make sure that the thread tool is cutting on both sides of the 
thread when at the proper length of worm. This rule is important and if 
not followed the worm will have an error in the lead both at beginning and 
ending of thread. Third: Between the centers of the lathe, with special 
fixtures fastened to the carriage, finish the radius. Use the micrometer with 




FIG. 145. SPECIAL FIXTURE CUTTING WORMS OR HOBS. 

double ball points to measure the diameter at the center of the radii or 
smallest diameter. Fourth: Strike a fine line around the piece at the 
smallest diameter. Note: Care must be taken to get this line accurately 
located, as all horizontal measurements are to be taken from this line. A 
special pointed tool is used for this operation. Fifth: Rough out the thread 
with special roughing tool. Sixth: Finish thread with the special thread 
tool. Seventh: Face off the ends of the threaded parts to a proper distance 
from the center line and finish-turn both bearings complete. Fig. 145 
shows the lathe set up with special fixture-cutting worms or hobs. 



MAKING THE HOB FOR CUTTING HINDLEY WORM-WHEELS 

"First Operation: Between centers of engine lathe rough out the blank 
of the hob, leaving the stock at each end of the part to be threaded equal to 



WORM GEARS 201 

not less than one-half the pitch. This is important as the end teeth of the 
hob must be equidistant from the center line and have full cutting surfaces. 
Second: Between centers in the lathe, with the special fixture, similar to that 
shown in Figs, i and 2, fastened on the carriage, cut the radius to finish, 
using the double ball-point micrometer for measuring the diameter at the 
smallest diameter. Third: Strike a line around the piece at the smallest 
diameter. Note: Care must be taken to get this line accurately located as 
all horizontal measurements are to be taken from it. Also, it is used in 
cutting the worm-wheel in proper relation to the hob when a located tooth 
is required. Fourth: Rough-out thread with special tool. Fifth: Finish 
thread with special thread tool. Note: The hob should be larger in diameter 
than the worm by one-tenth of the thickness of the tooth at the pitch line. 
Use the gage or template for depth and width of thread, also, micrometer 
with ball point, sleeve. Sixth: Strike a line parallel with the axis, crossing 
the line described in operation No. 3, in the exact center of the top of the 
thread. This line is essential when a located tooth in the worm-wheel is 
required. Seventh: Mill the flutes deep enough to establish a cutting edge 
at the bottom of the thread, as the hob must finish the face of the worm-wheel; 
also, care must be taken to have the cross lines come in the center of the 
tooth. Mill the flutes square with the helical angle of the thread. Eighth: 
Cut the hob to same length as the worm and remove any teeth on each end 
of hob which would be liable to break off while the hob is cutting. Care 
must be taken to have the same number of teeth on each end of the hob, 
counting from the one with the center line or cross-line. Ninth: Face the 
ends of the threaded part, measuring from the center line to get the faces an 
equal distance from the center of the radii. Tenth: Back off teeth. Note: 
Care must be taken in this operation to get an equal amount of clearance on 
each side of the tooth. Eleventh: Turn the bearing surfaces at each end of 
the threaded part. Allow about 0.015 inch for grinding after hardening the 
hob. Twelfth: Harden the hob. Note: Temper this hob only at the point 
of the teeth. Care must be taken that the bottom of the teeth is hardened 
as this part of the hob must form the diameter of the wheel. Thirteenth: 
After tempering, set the hob in the centers of the lathe and get the teeth 
running true. Then recenter each end and turn to finish. 

MAKING THE WORM-WHEEL 

" First Operation: Turn up the worm-wheel, leaving the outside diameter 
about 0.02 inch large, to be finished with the hob. Second: Mill the teeth in 
the wheel. Note: The exact centers of the hob must be in the center plane 



202 



AMERICAN MACHINIST GEAR BOOK 



of the worm-wheel when teeth are required to be in a fixed relation to some 
other part of the wheel or segment. Turn the hob around in the machine 



^ 




FIG. I46. MACHINE IN OPERATION. 

"^-#-#^g l -|§Hg| "H \% H| if M tn ti fci si ^ W\ 1 




FIG. 147. RACK, WORM AND HOB OF HINDLEY GEAR 

until the center of the tooth which has the center line is exactly on top, and 
then set the work in the machine, locating from the center of the cross-lines. 



WORM GEARS 203 

To get the proper depth of the tooth, measure with a double ball-point mi- 
crometer from the center of the bore to the center of the radii on top of the 
tooth. If the hob is accurately made and located, this measurement will 
be found reliable. Note: The worm-wheel must be driven at the proper 
lead by an independent set of gears. 

"The illustration, Fig. 146, shows a machine in operation with the gear 
segment being cut with a hob to the Hindley type of tooth. Fig. 147 shows 
a rack, worm and hob of the Hindley type of tooth. 

DIAMETRAL PITCH WORMS 

"If the proper change gears are provided, it is as easy to cut diametral pitch 
worm teeth as any. The proper gears can always be easily calculated by the 
rule that the screw gear is to the stud gear as 22 times the pitch of the lead 
screw of the lathe is to seven times the diametral pitch of the worm to be 
cut. For example, it is required to cut a worm of 12 diametral pitch, on a 
lathe having a leading screw cut six to the inch. We have: 

Screw gear 22 X 6 11 

Stud gear 7 X 12 7 ' 

and any change gears in the proportion of n and 7 will answer the purpose 
with an error of io So o" of an inch to the thread of the worm. If 22 and 7 
give inconvenient numbers of teeth, the numbers 69 and 22 can be used with 
sufficient accuracy, and 47 and 15, or even 25 and 8, may do in some cases." * 

Care should be taken when using these calculations that the same change 
gears are used to chase both the hob and the worm, as a slight difference in 
the lead of one tooth may prove a serious matter in a worm engaging a large 
gear where several teeth will be in contact. This same precaution applies 
to worms and hobs of a fractional circular pitch. 

It should also be remembered that 4 pitch does not mean 4 threads per 
inch measured on the axis of worm, but 4 threads per inch of diameter of 
the engaging worm gear. The corresponding circular pitch is 0.7854 inch, 
not 0.25 inch. 

*George B. Grant's Treatise on Gearing, Section 120. 



SECTION VII 
Helical and Herringbone Gears 

The exacting demands of smoothness and silence in operation, long life 
and high efficiency for high-speed gear transmission, such as those imposed 
upon the reduction gears for steam turbines, are simply met by helical or 
herringbone gears only. Such gears are superior to the. common spur gear 
in all the requirements made by this trying service. The obliquity of the 
teeth keeps one set in mesh until the following set of teeth is well engaged 
so that at no time is there a sudden transference of load from one tooth to 
the next, as occurs in ordinary spur gearing. The load is gradually put on 
a tooth and as gradually taken off so that the strain on the teeth is kept 
practically constant and the sudden shock of impact, common to spur 
gearing, is avoided. 

In the ordinary spur gear the teeth come in contact over their entire 
length at one time and the whole load is first thrown on the end of the tooth, 
producing the maximum leverage strain as soon as contact takes place. 
This leverage strain is subsequently reduced, but it takes place suddenly 
and, therefore, is much more serious than if led up to gradually. In gears 
with oblique teeth, the load is put on each tooth gradually and as gradually 
removed so that no severe leverage strain is created at any time. 

Helical gears, examples of which are diagrammatically depicted in Figs. 
148-153, possess the drawback of exerting a more or less serious axial or 
side thrust on account of the action of the teeth. This thrust varies with 
the angle of the spiral, so that the highest efficiency is obtained when the 
angle is only great enough to assure the accurate and gradual meshing of a 
set of teeth during the equally gradual releasing of the preceding set of teeth. 
The angle of the spiral need be such only that the end of one tooth will just 
overlap the end of the adjoining tooth. It follows, therefore, that the wider 
the face of a helical gear, the less the angle of spiral need be and the less the 
side thrust produced. 

When employing helical gears it is always desirable to use two gears of 
opposite pitch or spiral in the same shaft, so that the axial thrust of the two 
gears will balance. Ordinarily this is accomplished by the use of duplicate 
sets of gears similarly mounted on common shafts. Where space is limited, 

204 



HELICAL AND HERRINGBONE GEARS 



205 



gears of opposite spiral may be mounted side by side to form virtually one 
gear, such gear being actually a herringbone gear. The exact balancing of 
the helical gear thrust may not always be feasible, nor may it be advisable to 
employ herringbone gears, and in such instances partial balance of thrust 




Fig. 148 






Angle of Spiral 





M 1 v\ 

MM 




J 




Fig. 149 


Circular Pitch v 

1 /Normal Pitch 




W/ZM/lll 





rig. 151 



1 



5[* 



Fig. 150 




Fig. 152 
Normal Circular Pitch 



Circular Pitch 




Circular Pitch 



Fig. 153 
THE DESIGN OF HELICAL GEARS. 



may often be realized. Fig. 150 illustrates such a case; the thrust on shaft 
B is balanced, and that on shafts A and C is reduced to a minimum. 

Two circular pitches are employed for helical gearing: the "normal 
circular pitch," which is the shortest distance between the centers of con- 



206 AMERICAN MACHINIST GEAR BOOK 

secutive teeth and is measured on an imaginary pitch cylinder, and the 
" circular pitch," which is the distance between the center of two teeth 
in the plane of the gear and which is measured on the pitch circle as for spur 
gears. See Figs. 151, 152 and 153. 

The pitch diameters must be in proportion to the number of teeth and 
both the circular and normal pitches must be the same in both gears of a 
pair. The angle of spiral must also be the same but of opposite hand. 

The form of tooth employed for helical and herringbone gears is the involute 
or a close approximation of that form. 

NOTATION FOR HELICAL AND HERRINGBONE GEARS 

D f = pitch diameter. 

D = outside diameter. 

N — number of teeth. 

p = diametral pitch. 

p' = circular pitch. 

p' n = normal circular pitch. 

E = angle of spiral. 

s = addendum. 

/ = clearance. 

W = whole depth of tooth. 

t = thickness of tooth. 

L = lead. 

B = center distance ) 

V . . } pair of gears. 

— = speed ratio 



FORMULAS FOR HELICAL AND HERRINGBONE GEARS 

v I for gear, (i - ^) 




— Xi 

= 2 I V ) for pinion. (i - #) 

, T 3.I4l6Z>' , V 

N=*-*t, (2) 

P 

D= D' + 0.6366 p'\ (3) 



HELICAL AND HERRINGBONE GEARS 207 

CosE= V -j' (4) 

p 

^ 3.1416 . . 

t>'n= p' COS E = — j (5) 



p'n 

:os J 

3.I4I6Z)' 



P' = —rJ (6) 

r cos E 



N 



(6-a) 



* 3- I 4 I 6 an 

P = ~~^r' w) 

s= 0.3183 *'». (8) 

W = 0.6866 p' n . (9) 

L= 3.1416 Z) r co/ E. (11) 

The corresponding spur cutter =- ( ^-3* (12) 

DISCUSSION OF FORMULAS 

Only the pitch diameter and the number of teeth are figured from the 
circular pitch. 

The outside diameter and all tooth parts are figured from the normal 
pitch, the relationship existing to the normal pitch being similar to the 
relationships of spur gears to their circular pitchs. 

The angle of the spiral, in practice, is usually selected with a view to the 
spur cutter available. The normal pitch divided by the circular pitch 
gives the cosine of the angle of the spiral. This angle should be kept as low 
as possible, ordinarily not exceeding 20 degrees, in order to avoid undue axial 
thrust. When the angle is fixed as well as the pitch of the cutter, the di- 
ameter necessary to give the proper combination may be found by first 
determining the circular pitch. A change in the angle to accommodate an 
even number of teeth makes no difference in helical gears. 

The angle of the spiral must be accurately adhered to, for a slight deviation 
will interfere with the proper contact of the teeth. 

The normal pitch should conform as nearly as possible with some standard 
pitch, so it is customary to assume the normal pitch and proportion the angle 
of spiral accordingly. If a variation between the pitch of the cutter and the 
normal pitch cannot be avoided, it is better to have the cutter pitch finer 
rather than coarser than the normal pitch. 



208 



AMERICAN MACHINIST GEAR BOOK 



The circular pitch, found by dividing the normal pitch by the cosine of 
angle of spiral, must always correspond to an even number of teeth — i.e., 
the product of the circular pitch and the number of teeth must represent 
the pitch circumference of the gear in question. 

After settling the pitch of the cutter, the number of teeth for which the 
cutter should be made is found by dividing the actual number of teeth in 
the gear by the third power of the cosine of the spiral angle. 

The spiral lead is the distance traveled by the thread in one complete 
revolution of the pitch circle. As the angle of spiral becomes smaller and the 
form of the tooth approaches that of a spur gear, the lead becomes longer 
until, when the spiral angle is zero, it lengthens to infinity. 

EXAMPLES IN THE DESIGN OF HELICAL GEARS 

i. Required: 

Pair of helical gears cut with a 4-pitch cutter, speed ratio '4 to 1 and center 
distance 12^2 inches. 

4-pitch cutter = 0.7854 inch circular pitch (Table I), 

= normal pitch for helical gears. 
Solution: 



DIMENSION 



FORMULA 



GEAR 



PINION 



Pitch diameter 

Number of teeth 

Normal pitch 

Circular pitch 

Angle 

Addendum 

Whole depth of tooth. 
Thickness of tooth 

Outside diameter 

Lead 



Cutter used. 
Hand 



(i-a and b) 

(2) 

Table I 

(6-a) 

Cosine from (4) 

(8) 

(9) 
(10) 

(3) 
(") 



20 . 000 

76 
0.7854" 
0.8267" 

18 n' 
0.250' 
Q-539' 
0.39 2 ' 

20. 500' 
191 . 292' 

No. 2-4P. 
R. H. 



5 . 000" 

J 9 

0.7854" 
0.8267" 
18 11' 
0.250" 

o.539" 

0.392" 

5-5oo" 

47.823" 

No. 5K-4P- 
L. H. 



2. Required: 

Pair of helical gears to replace two spur gears of 60 and 1 5 teeth respectively, 
3 pitch, 12^-inch centers. Speed ratio to remain the same and the helical 



HELICAL AND HERRINGBONE GEARS 



209 



gears to be cut with the same pitch cutter but to have fewer teeth in order 
not to change the center distance. 

3-pitch cutter = 1.0472 circular pitch (Table I), 

= normal pitch for helical gears. 
Solution: 



Dimension 



formula 



GEAR 



PINION 



Pitch diameter 

Number of teeth 

Normal pitch 

Circular pitch 

Angle 

Addendum 

Whole depth of tooth. 
Thickness of tooth. . . 

Outside* diameter 

Lead 



(i-a and b) 

Assumed 

Table I 

(6-fl) 

Cosine from (4) 

(8) 

(9) 
(10) 

(3) 
(11) 



Cutter used 
Hand 



20.000 

56 
1 .0472' 
1 . 1220' 

21 3 

O.S333' 

O.9075' 

0.5236' 

20.666" 

I63.4S6" 

No. 2~3p. 
R. H. 



5 . 000" 

14 
1 .0472" 
1 .1220" 

21° 3' 

0.3333" 
O.9075" 
O.5236' 
5.666'" 
40.864" 

No. 7-3P. 
L. H. 



When space permits or when helical gears can be arranged in pairs so as 
to overcome the axial thrust that constitutes the chief drawback of these 
highly efficient gears, they are usually to be preferred to herringbone gears. 




FIG. 154. HELICAL GEARS FOR CUTTER DRIVE. 

They are easier to machine and possess the always desirable feature of 
greater simplicity. 

A well-balanced helical gear drive is shown in Fig. 154, which illustrates 



210 AMERICAN MACHINIST GEAR BOOK 

the arrangement of gears on a heavy rack cutter. Before the installation 
of these gears, ordinary spur gears had been employed with discouraging 
results. The spurs had broken, worn out rapidly and had proved entirely 
inadequate for the service demanded. The helical gears, on the other hand, 
have proved eminently satisfactory. After several years of operation they 
show no appreciable wear, run smoothly and with little noise, and permit the 
rack cutter to be driven at far higher speed than was formerly possible. 

Helical speed-reduction gears have also proved very successful and some 
remarkably high efficiencies are reported for such apparatus. The De 
Laval Steam Turbine Co., which has adopted such mechanism in connection 
with its steam turbines, claims efficiencies as high as 99 per cent, and states 
that an efficiency of 98^2 per cent, is a conservative figure. This enviable 
record being only possible through excellent design, the use of proper materials 
and high-grade workmanship, a brief description of this gear will be of 
interest. 

DE LAVAL SPEED REDUCTION GEAR 

Two helical gears with their pinions are used to overcome the axial thrust 
and to permit a somewhat greater spiral angle to be employed than is custom- 
ary. The angle of spiral is such that several teeth are in contact at the same 
time. The form of tooth used is the involute to secure true line contact as 
long as the tooth is in mesh. 

The gears consist of a rigid cast-iron center or spider upon which steel 
bands of a special grade of steel are shrunk. The gear blanks are carefully 
mounted and trued up before the teeth are cut in order to insure accuracy. 

The pinion is cut directly on the pinion shaft, which is a special nickel- 
steel forging that is oil-tempered to the desired degree — the pinion having 
to be considerably harder than the gear bands to assure uniform wear and 
long life. 

After the gear and pinion have been cut, they are carefully polished to 
remove any tool marks and unevenness. This is accomplished by running 
the gear and pinion with similar and as accurately cut "dummy" gears." 
This polishing process naturally adds to the cost of manufacture, as the 
"dummy" gears rapidly deteriorate from wear and have to be discarded. 
The added cost of polishing is well warranted, however, as it greatly increases 
the life and efficiency of the finished gear. 

The bearings, lubrication and details of the gear case for such speed re- 
ducers are not dissimilar to those of any other high-grade speed-reduction 
gear. 

Special flexible couplings for connecting the pinion shaft to the steam tur- 



HELICAL AND HERRINGBONE GEARS 



211 



bine and the gear shaft to the machine to be driven are necessary, of course, 
for any mis-alignment would be suicidal to the high efficiency demanded. 
Speed reduction gears built with this care have been opened up after more 
than 10 years of constant and exacting service and have shown no wear. 



HERRINGBONE GEARS 

A double-helical or herringbone gear avoids the axial thrust of the single- 
helical gear which is not properly balanced, consisting as it does of virtually 
two gears exerting axial thrusts in opposite directions and thereby nulli- 
fying any unbalanced side pressure. 

The formulas for herringbone gears are the same as for helical gears, but 
the angle of spiral can be made considerably more obtuse. In fact, the angle 
of the teeth with the axis of the gear is only limited by constructional diffi- 
culties. Angles of 30 degrees are quite common and angles as great as 45 
degrees are quite frequently used. 







mm 



■■'/'//Icv'S/A 



FIG. I5S. 



I 



FIG. 156- FIG. 157. 



THE DESIGN OF HERRINGBONE GEARS. 




Fig. 155 shows a common type of herringbone gear, and Figs. 156 and 157, 
modifications that are frequently resorted to to facilitate manufacture. The 
gear shown in Fig. 155 really consists of two helical gears fastened together. 
The gears are cut separately and not connected until after machining. This 
simplifies the cutting of the teeth but also adds to the cost of manufacture. 
The type shown in Fig. 1 56 differs only in that the rim is made in two pieces 
and separate from the spider. The same, ease in cutting the teeth is thus 
secured as in the other type and the added advantage of simplifying replace- 
ments should the gear teeth be damaged or wear out. 

The grooving of the type shown in Fig. 157 allows the gear to be made 



212 



AMERICAN MACHINIST GEAR BOOK 



in one piece, as the cutter can be run out at the groove. In this type of con- 
struction, the teeth may be staggered to somewhat lessen the width of the 
central groove. 

The great difficulty of accurately cutting herringbone gears so that they 
may be interchangeable has led to the adoption of many makeshifts. When 
the gear is made in one piece, the pinion is sometimes made in two pieces to 



JE 



* 




FIG. 153. ARRANGEMENT OP PINION. 

facilitate the proper engagement of the teeth. The pinion is run with the 
gear and not keyseated until after the teeth have accurately adjusted them- 
selves to the position of the teeth in the gear. 

Another plan that was suggested by "Attic" in the American Machinist 
consists of placing a washer of some elastic material between the two halves 
of the pinion, as shown in Fig. 158. so that they can adjust themselves by a 
slight axial movement, a movement along the shaft having the same result 
as if the hah gears were slightly turned about the shaft. 

INTERCHANGE ABLE SYSTEM FOR HERRINGBONE GEARS 

Percy C. Day presented an interchangeable standard for herringbone gears 
before the American Society of Mechanical Engineers. 

The proposed standard, which has been adopted by at least two important 
manufacturers of herringbone gears, is as follows : 

Tooth shape Involute. 

Pressure angle 20 degrees. 

Spiral angle 23 degrees. 

-r,. , .. , . N number of teeth 

Pitch diameter (20 teeth and over) = -j-. ; — = — r- 

diametral pitch 

_..... . . , . number of teeth — 1.6 

Blank diameter ( 20 teeth and over) = y- i — : — Z 

diametral pitch 

w , ,. , , 0.95 X number of teeth + 1 

Pitch diameter (under 20 teeth) = — — — j-- i — = — r ' 

diametral pitch 

m , ,. , , , . 0.95 X number of teeth +2.6 

Blank diameter (under 20 teeth) = ~ ; — : — r ■ 

diametral pitch 



HELICAL AND HERRINGBONE GEARS 213 

Addendum * p • 

_ 1.0 

Dedendum tTp~' 

1 8 
Full tooth depth ' p • 

Working tooth depth tTp - ' 

Standard face width for gears with pinions of not less than 25 teeth, six 
times the circular pitch. 

Face widths for high-ratio gears with small pinions, six to twelve times 
the circular pitch. 

When a pinion of less than 20 teeth is used with a standard gear, the 
center distance must be slightly increased to suit the enlargement of the 
pinion. If it is desired to keep the center distance to the standard dimen- 
sions, the gear diameter may be reduced by the amount of the enlargement 
given to the pinion. 

For example: If a pinion of 10 teeth, 5 diametral pitch (D.P.) is to mesh 
with a gear of 90 teeth at 10-inch centers: 

Pitch diameter of pinion =2.1 inches. 

Enlargement over standard pinion = 0.1 inch. 

Pitch diameter of standard gear = — = 18 inches. 

Reduced pitch diameter of gear = 18 — 0.1 = 17. 9 inches. 

Center distance — ' — = 10 inches. 

2 

Strictly speaking, there can be no enlargement for reduction of the pitch 
diameter in a pinion or gear of given pitch and number of teeth. It is con- 
venient to assume such enlargement or reduction, however, when using teeth 
of long and short addenda but standard depth. 

MACHINING HERRINGBONE GEARS 

Three general methods of cutting herringbone gear teeth are in vogue: 
(1) A double-hobbing machine of special design is employed by which the 
right- and left-hand teeth are cut simultaneously. In such machines the 
cutting profile of the hob teeth, cutting their way into the gear blank on a 
diagonal line conforming to the obliquity of the teeth, are modified from the 
true involute form in order that the space cut may closely approximate the 



214 



AMERICAN MACHINIST GEAR BOOK 



correct involute profile in the direction of rotation. For extreme accuracy, 
therefore, a uniform obliquity of tooth is necessary, or else a special hob for 
each angularity of tooth. (2) The method second in importance is that of 
planing the teeth. Planers are employed which use a cutting tool having 
the same section and form as the finished tooth space, or machines with 
templets for guiding simple planing tools are used. (3) Herringbone gears 
of the smaller sizes are also cut with single rotary cutters. 

HOBBIXG PROCESS TOR CUTTING HERRINGBONE GEARS 

Machines of two general types are employed in the hobbing process. The 
working principle of the type shown diagrammatically in Fig. 159-tf is as 
follows : 

The gear blank a is mounted on the vertical mandrel c, which is rotated by 
the face plate b through the spindle d actuated by the driving worm gear e. 




FIG. 159-d. DIAGRAM OP HOBBIXG PROCESS 
FOR CUTTING DOUBLE HELICAL GEARS. 



The hobs // are mounted in vertical slides g g. which move up and down on 
the standards h h. These standards are mounted so as to slide on the bed of 
the machine 7 and are provided with micrometer screws for adjusting the 
depth of cut. 

The rotating speed of the gear blank is controlled by a train of wheels 
which operate a dirterential gear so as to give the required spiral lead to the 
teeth. The process is entirely automatic and evades the necessity of in- 
clining the hob axes, the lead being governed by the speed at which the gear 
blank is revolved. This allows the hobs to be set at right angles to the axis 
of the gear blank and the same hobs to be used for gears of difierent obliquity 
of tooth, provided extreme accuracy is not essential. 

Fig. 159-3 shows the other type of hobbing machine for cutting herring- 
bone gear teeth. This single-headed machine has two hobs, a right- and a 
left-hand one, carried on a single saddle. The hobs rotate in opposite 



HELICAL AND HERRINGBONE GEARS 



215 




FIG. I59-6. HOBBING MACHINE FOR CUTTING DOUBLE HELICAL GEARS. 




FIG. I59-C. MACHINE FOR PLANING DOUBLE HELICAL GEARS. 



216 AMERICAN MACHINIST GEAR BOOK 

directions and are fed downward, cutting both halves of the gear at the same 
time. The cutting pressures are opposed and neutralized through the 
section of the gear being cut, thus relieving the machine of any such un- 
balanced stresses. 

The lower hob is carried on a spindle, while the upper hob is carried on a 
slide which is vertically adjustable. This allows the hobs to be set for cutting 
gears of various widths, and the adjustable features of the hobs permit them 
to be set so as to cut teeth with apexes on the center line of the gear or teeth 
of the staggered variety. 

The desired depth of cut is fixed by horizontal adjustment of the work 
carriage, after which the cutting operations require little attention, the 
retardation and acceleration of the hobs, the rotation of the work table or 
carriage, etc., being entirely automatic. 

PLANING PROCESSES FOR CUTTING HERRINGBONE GEARS 

Two general schemes of planing herringbone gear teeth are employed: 
first, that in which a tool of the exact section and form of the space to be cut 
is used; and second, the one in which templets are employed to guide the 
simple cutting tools employed — the "former" method. The first method is 
only employed for cutting comparatively small gears with circular pitches 
usually under i inch, the second for machining larger gears of coarser 
pitch. 

A large herringbone gear planer in which the tooth profiles are shaped by 
formers is shown in Fig. 159-c. The two tool saddles move toward each 
other on the cutting stroke, both halves of the gear being cut at the same 
time by tools advancing from the outer edges of the gear to the center apex 
line of the tooth. The cutting pressures of the tools are thus resisted by the 
gear itself and neutralized, instead of being taken up by some part of the 
machine. The main shaft carries two worm-wheels, the larger for indexing 
and the smaller for rotating the gear blank. Two sets of change gears, one 
for the angle of the tooth and the other for the number of teeth, together 
with a reversing drive constitute the special features of this particular planer. 

MILLING PROCESS FOR CUTTING HERRINGBONE GEARS 

The following description of the milling process for cutting herringbone 
gear teeth is taken from an article in American Machinist by Percy C. Day, 
as is also the test under the three following sub-titles. 

" In the milling process the teeth are sometimes cut by means of end mills 



HELICAL AND HERRINGBONE GEARS 



217 



formed to the tooth shape on the normal section. The working principle of the 
machines usually employed is shown in the diagram, Fig. 160. The end mill a is 
supported by the saddle b, which traverses the bed c. The mill is driven by 
the bevel gears d from the splined shaft e and driving cone /. The feed and 
differential motions are driven from e through speed cones or gears g and 
clutch h. The traverse of the saddle b is actuated by the feed screw j. Motion 
is also transmitted from/ through change wheels k, reversing gears /, dividing 
change wheels m, worm 0, dividing wheel w, and work spindle p to the blank q. 





FIG. 160. DIAGRAM OF MILLING PROCESS 
OF CUTTING DOUBLE HELICA*L GEARS. 



a = Half Circumferential Pitch o ==a - 

b = Spaee cleared by Cutter at the Turn 

The Bhaded Portion must be removed before the 
Teeth will Mesh 

FIG. l6l. DIAGRAM OF SPACE NOT CUT 
BY MILLING PROCESS OF CUTTING 
DOUBLE HELICAL GEARS. 



" While the end mill traverses from one edge of the blank to its center, the 
blank is rotated through an angle which gives the requisite spiral form to the 
tooth. The saddle then operates a stop which is in connection with the 
reversing gear /, and the rotation of the blank is reversed until the end of the 
cut. A quick-return mechanism, not shown in the diagram, comes into action 
at the end of the cut, and the mill is returned to the starting position. The 
dividing mechanism m is then operated by hand, and the cutting process is 
repeated on another tooth. 

"The end-milling process can be readily adapted for cutting double helical 
bevel gears. 

"The disadvantages of the process are principally of a practical nature. 
End mills are small tools, and are liable to rapid wear. Since the teeth are cut 
singly, any wear on the mill causes a change of tooth shape and thickness. 
The reversal of the angular motion of the blank while cutting proceeds allows 
the inevitable blacklash in the mechanism to take effect in a manner which 
is not conducive to accurate work. The cutter must be formed to the normal 
tooth section, and has not the circumferential shape of the teeth which it cuts. 



2l8 AMERICAN MACHINIST GEAR BOOK 

The width of the tooth space at the apex corresponds to the normal instead 
of to the circumferential pitch, hence the space must be cleared out by 
hand or in a separate operation (see Fig. 161). 

"The tendency to wear is greater when the end mills are small, and wheels 
on this system are generally made of coarser pitch than is really necessary or 
even desirable from the user's point of view, in order to minimize the manu- 
facturing difficulties by the use of large mills. 

ADVANTAGES OF THE DOUBLE HELICAL SYSTEM 

"The adoption of the double helical principle in gearing, if properly applied, 
reduces noise to a minimum and practically ehminates vibration without any 
necessity for departure from sound mechanical principles. In this type of 
gear, pinions may be chosen of sufficient hardness to wear evenly with the 
wheels, and soft materials do not enter the proposition. This is due to the 
absolute continuity of engagement which is characteristic of double helical 
gears when accurately cut and correctly designed to suit the working condi- 
tions. The effect of vibration is not by any means confined to the gears 
themselves, but acts injuriously on the shafts and machinery connected 
therewith. Many failures of haulage and other gear-driven shafts have been 
directly traced to this cause. 

"Consider a pair of wheels transmitting ioo horse-power with an efficiency 
of 96 per cent. If we assume only one-tenth of the lost energy to be dissipated 
in vibration which is absorbed in the wheel shaft, the result is somewhat 
surprising. Under such conditions the shaft is called upon to absorb energy 
at the rate of nearly eight million foot pounds during each working day of 
10 hours duration. The result is finally expressed in crystallization of the 
shaft material. . . . 

"Another interesting application of machine-cut, double helical gears is 
the reduction of speed from high-power steam turbines. No other type of 
gear can be used for this class of work, because absolute smoothness of action 
is essential. The essence of this problem is to avoid excessive velocity by 
keeping the pinion diameter small, but at the same time it is undesirable to 
reduce the number of teeth below a certain point because absolute continuity 
of engagement must be maintained. The result of these conditions is that 
the gears must be of extremely fine pitch and great relative width. For 
example, a set of gears recently constructed for a 500-horse-power steam tur- 
bine, to reduce from 3,000 to 300 revolutions per minute, were 4 diametral 
pitch with face width 10 inches and pinion of 19 teeth. 

"There is probably no field of application for double helical gears which 
offers such substantial advantages as for driving machine tools. In most 



HELICAL AND HERRINGBONE GEARS 



219 



modern machine shops there is a tendency to dispense with shafting as far as 
possible, and to drive the tools individually from separate motors. This 
method allows a more economical distribution of machines over the available 
floor space, and leaves the space overhead clear for rapid handling. On the 
other hand, motor-driven tools require far more gearing than when the drive 
is effected by belts, and it has been found difficult to obtain uniformity of 
motion under the new conditions. If 
machine-cut, double helical gears are 
used for this purpose, the quality of 



P a =1.0fc 



»Ie=l 




t R =l 






'<S=1 



Ph 



■esS? 




.3> 




the work turned out is much improved 
and, by reason of reduced vibration, 
higher speeds and coarser feeds can be 
employed. 

"The diagram, Fig. 162, shows the 
relative normal tooth pressures, pitch 
line- sections and stresses for angles of 
23, 45, and 60 degrees. 

"One of the greatest advantages of 
machine-cut, double helical wheels is to 
be found in their adaptability for high 
ratios of reduction. The number of 
teeth which can be used with success 
in the smallest pinion lies far below 
the practical limit for straight spurs, 
and pinions of four or five teeth are 
by no means uncommon for special 
purposes. Since, however, the pitch 
can be made very fine, it is rarely 
necessary to reduce the number of teeth 
so far, and most high-ratio gears are 
made with pinions of 11 to 20 teeth. 
Pinions for high ratios are generally cut 

solid on their shafts, in order that the diameters may be kept low to bring the 
wheels within reasonable proportions. 

" Single wheels and pinions will transmit heavy powers with ratios between 
10 and 20 to 1, so that they can be used in place of worm gears or double 
trains of ordinary spurs. As against worm gears the gain lies in the direction 
of increased efficiency and life. A set of double helical gears with 20 to 1 
ratio has an efficiency of about 95 per cent, against a maximum of about 80 
per cent, for a worm gear of equal ratio. 




p tt =i 



P = Circumferential Tooth Pressure 

P n = Normal Pressure 

P a = Axial Pressure 

t e = Circumferential Tooth Thickness 

i n = Normal Thickness 

f c = Circumferential Stress 

f n — Normal Stress 

FIG. 162. COMPARISON OF TOOTH PRES- 
SURE, THICKNESS, AND RELATIVE STRESS 
FOR THE TOOTH ANGLES OF DOUBLE 
HELICAL GEARS. 



220 



AMERICAN MACHINIST GEAR BOOK 



AN INTERESTING APPLICATION OF DOUBLE HELICAL GEARING 

"An interesting example of this difference came under my notice a short 
time ago. A worm gear of first-class manufacture and modern design had 
been in use for some 2^/2 years, driving a deep-well pump from a 50-horse- 

power motor with reduction 480 to 22 
revolutions per minute. This gear was re- 
placed by a double train of machine-cut, 
double helical wheels, the ratios being 480 
to 60, and 60 to 22. The records of power 
consumption and pump duty were regu- 
larly kept, and after the new gear had 
been running for a year the figures showed 
a net saving of over 17 per cent, in its 
favor as against the average for the whole 
life of the worm gear. It was also shown 
that the efficiency of the double helical 
gear had actually improved after a year's 
daily work, while the worm gear had 
steadily deteriorated in this respect from 
the day it was started. 

"Fig. 163 shows a set of double helical 
gears that are representatives of their de- 
sign and construction. 



IMPORTANT POINTS IN APPLYING DOUBLE 
HELICAL GEARS 



" In conclusion it is desirable to add a 
word of caution to those who are about to 
adopt this class of gear for the first time. 
It must not be forgotten that there are 

three fundamental points of difference between machine-cut, double helical 

wheels and ordinary spur gearing: 

" (a) The pitch is finer. 

" (b) The face width is greater. 

" (c) The tooth pressures are generally higher. 

"To insure satisfactory working it is necessary that the shafts shall be 
parallel, true, and rigidly supported. The center distance must also be ad- 
justed with great care on account of the fine pitch and small clearances allowed. 




FIG. 163. TYPICAL HIGH-RATIO 
GEARS WITH STAGGERED TEETH. 



HELICAL AND HERRINGBONE GEARS 



221 



Motor pinions of high-ratio gears should be mounted on extended shafts 
with an outer bearing. Anything in the nature of an overhung drive should 
be avoided wherever possible. 

"To avoid undue wear from magnetically controlled end- thrust in motors, 
the pinions should be mounted on two parallel feathers set at 180 degrees 
and carefully bedded to the keyways 
(see Fig. 164). The pinions should be a 
good tight fit on the motor shafts, but 
there should be just sufficient freedom to 
allow them to move along under the in- 
fluence of continued side pressure, so that 
the motor armature can reach a neutral 
position where the pressure ceases. It is 
unnecessary to allow the pinions to slide freely on the shafts, and if this is 
done there may be trouble from excessive wear of the keys and keybeds." 





y 










i_ 


H 








i_ 










a 






FIG. 164. AN IMPROVED METHOD OF 
KEYING FOR GEARS. 



DETERMINING LEAD AND ANGLE FROM SAMPLE 

To produce a herringbone gear to operate with a sample, the calculations 
for which are unknown, is generally a matter of cutting and trying until a 

satisfactory gear is produced, as for 
herringbone or helical gears the angle 
and lead must be exceptionally ac- 
curate, the teeth having contact 
their entire length, and a slight 
error is noticeable. There is more 
or less leeway for spiral gears, but 
the method as described can be 
applied to them as well. 

Cover the points of the teeth in 
sample with an application of lamp- 
black, or anything that will make 
a clear impression on a piece of 
clean white paper. Roll the gear 
thus treated on the surface of the 
paper, being careful not to allow 
it to slip, until a sharp impression 
of the points of the teeth is made, 
as illustrated in Fig. 165. This will represent a development of the teeth at 
the outside circumference. 

The angle of the teeth at the outside circumference may then be measured 




FIG. 165. IMPRESSION MADE BY ROLLING 
SAMPLE HERRINGBONE GEAR. 



222 



AMERICAN MACHINIST GEAR BOOK 



/ 



with a protractor by extending the lines of the tooth as developed on the 
paper. 

= Face of herringbone gear. 

= Length of the tooth from center of face. 

= Angle of spiral at outside diameter. 

= Angle of spiral at pitch diameter. 

= Lead of spiral. 

= Outside circumference. 

= Pitch circumference. 



p 

L 

c 



cos /^ = 



0.5/ 



For helical gears this formula would be: 

f 

cos /3 1 = — 



The next step is to find the lead: 



L = 



Ci 



tan B\ 

As the lead is necessarily the same at the outside diameter as it is at the pitch 
diameter of a helical or spiral gear when cut with a rotary cutter, the angle 
of spiral at the pitch line may be found by formula 4. 

C 

Tan (3 = — . 

The fact that the lead is the same at all points when cutting a spiral, helical, 
or herringbone gear cutter, using a single rotary cutter, makes the solution 

of this problem a simple 
matter. Fig. 166 is self- 
explanatory. 

This being the case, it is 
apparent that such a cutter 
cannot reproduce its own 
shape in the gear blank, as 
to do this the angle and 
lead must be proportional 
to all parts of the tooth. 
When the teeth are gener- 




FIG. l66. 



American Machinist 
DIAGRAM OF ANGLES OF HERRINGBONE GEAR. 



ated this condition is ful- 
filled and the angle at the pitch line will be proportional to the pitch and 
outside circumferences, or: 



fi- 



c, 



HELICAL AND HERRINGBONE GEARS 223 

EFFICIENCY AND STRENGTH OF HERRINGBONE GEARS 

The efficiency of accurately cut herringbone gears for ratios up to 10 to 1 
is about 98 per cent. For greater speed ratios, their efficiency shows a 
slight falling off, but if a single reduction is not abnormal and the gear and 
pinion well mated, the efficiency should not be less than 96 per cent. Even 
better records have been realized by gears which have been particularly 
accurately cut, polished and run in an oil bath. Frequently an efficiency of 
99 per cent, or even slightly higher has been obtained by such gears with 
speed ratios not exceeding 10 to 1. 

The life of herringbone gears is far greater than that of even the most 
carefully cut spur gears, and though accurate data are hard to find on this 
subject, it may be safely stated that the usual life of a carefully cut herring- 
bone gear is at least three or four times that of a similar spur gear. 

W. C. Bates, Mechanical Engineer, Fawcus Machine Co., prepared for 
American Machinist a comprehensive article on the design and strength of 
herringbone gears, from which the following important points are abstracted. 

Due to the advantages of the herringbone construction, compared to that 
of spur gears, the load is always distributed over more than one tooth, the 
transference of load from tooth to tooth is without shock, the bearing 
pressure angularly placed on the tooth diminishes the strain on the root of 
the tooth, the tooth is shorter and therefore more sturdy, and the spacing 
of the teeth and tooth form are more accurate on account of the hobbing 
process employed in cutting the teeth, so that the indeterminate tooth 
stresses are at a minimum when running at high speeds. 

Such advantages naturally allow considerable modification of the well- 
known Lewis formula for the strength of gears, as such empirical formula 
pertains to the strength of herringbone gears. 

It then becomes: 

1200 



S = s' 



1200 + v 



where 



S = allowable stress per square inch at a speed of v feet per minute. 
s' = static stress = 8,000 pounds per square inch for cast iron, 

= 20,000 pounds per square inch for steel. 
v = speed in feet per minute. 

MODIFIED HERRINGBONE GEARS 

An efficient modification of the standard herringbone gear may be con- 
structed with a form of tooth that relies only upon its rolling action and 



224 AMERICAN MACHINIST GEAR BOOK 

eliminates all sliding contact. A tooth of the regular involute form on the 
pitch line is employed with the surplus metal above and below this contact 
line cut away. Such a gear will run as smoothly as the standard herringbone 
with full, tooth section as long as no appreciable wear takes place. This 
naturally limits the practical value of the type, but it is of interest as repre- 
senting the ideal in gearing action. 

A radical departure from any ordinary type of gear has recently been put 
on the market by the R. D. Nuttall Co., Pittsburg, Pa., which has been 
designated as "The Circular Herringbone." A description of this gear 
appeared in American Machinist, Oct. 16, 1913, from which the following 
excerpts are taken. 

"It has a continuous tooth curved across the gear face, the curve being a 
circular arc. This approximates the shape of a herringbone tooth, hence 
the name "The Circular Herringbone." The tooth profiles at the middle of 
the face are true involutes; other profiles vary slightly from this, but are 
close approximations to the involute form. Though any tooth proportions 
can be cut, those standardized are: A pressure angle of 20 degrees, addendum 
0.25, dedendum 0.25, clearance 0.05, working depth 0.5, and whole depth 
0.55 of circular pitch." 

These gears are generated, the cutter and blank rolling together during 
the process of cutting in such a manner that the line of tangency is along 
the pitch line of the cutter and the pitch surface of the blank. 



SECTION VIII 

Spiral Gears 

Before going into the matter of calculations it may be well to direct the 
readers to a careful consideration of the accompanying perspective sketches 
originally published in American Machinist, October n, 1906, by H. B. 




Follower 
Left Hand 
C Spiral 



Fig. 169 



Fig. 170 



Fig. 171 



SPIRAL-GEAR DIAGRAMS. 



McCabe. "In Figs. 167 to 171, inclusive, the driving gear of each pair 
is shown as if transparent, the teeth being represented by lines. Fig. 167 
shows a pair of gears on shafts at right angles and Fig. 168 a pair on parallel 

2 2S 



226 AMERICAN MACHINIST GEAR BOOK 

shafts. Note that in Fig. 167 both spirals are left hand, while in Fig. 168 one 
is left and the other right hand ; that is, in the first case they are the same hand 
and in the second case they are opposite hands. (The word hand as here 
used has the same significance as in the case of threads.) It is evident that 
these two are fixed conditions for shafts respectively at 90 degrees and parallel. 

"Now when the shafts are at any angle between 90 degrees and o degrees 
either of these conditions may exist; that is, the spirals may be both the same 
hand or they may be opposite hands. This may be made plain by observing 
carefully Figs. 169, 170, and 171, in which the shafts are at an acute angle, 
all conditions in the three views being exactly alike except that the teeth are 
at different spiral angles in each. Note that in Fig. -169 the spiral angle of 
the driver is the same as the angle of the shafts which makes the follower 
a plain spur gear. Also note that the spiral of the driver is left hand. Now 
letting the spiral of the driver remain left hand, but increasing its angle a 
little we have the condition of Fig. 170. By decreasing it a little we have 
the condition in Fig. 171, making in the first case the spirals opposite hands 
and in the second case the spirals the same hand. 

The lines A and O B in these figures are drawn parallel to the shafts and 
the line C is drawn tangent to the spiral of the teeth and makes with A 
and B respectively the spiral angles of the driver and of the follower. Note 
that in Fig. 171 the angle of the shafts A B equals the spiral angles AOC + 
B C, and in Fig. 170 the same angle A O B equals A O C — B C. 

RELATION OF SHAFT AXD SPIRAL ANGLES 

"The following general rules are now evident: 

"1. When the spirals are the same hand the angle of the shafts is the sum of 
the spiral angles. 

"2. When the spirals are opposite hands the angle of the shafts is the differ- 
ence of the spiral angles. 

"3. When the spiral angle of one gear is the same as the angle of the shaft 
the spiral angle of the other will be zero, making it a plain spur gear. 

"4. When the shafts are at right angles the spirals must both be the same 
hand. 

' '5. When the shafts are parallel the spirals must be opposite hands. (Helical 
gears.) 

"6. When the shafts are at any acute angle the spirals may be either the same 
hand or opposite hands." 

The following is an extract from an article on spiral gearing originally 
published in American Machinist by F. A. Halsey: 



SPIRAL GEARS 



227 



" Spiral gears are not to blame for the undoubted fact that they are somewhat 
troublesome to lay out, the difficulties of the problem being due to the limita- 
tions of workshop facilities and not to the geometrical nature of the gears 
themselves. It is easy to understand and explain the action of an existing 
pair of spiral gears. More than this, it is easy to lay out a pair of such gears 
which shall exactly meet all the conditions of the case except one; they cannot, 
except through rare good luck, be made with the appliances at hand. To be 
more specific, the circumference cannot usually be divided into an exact 
whole number of teeth by any stock cutter, and the real problem becomes the 




fig. 172. 




readjusting of the diameters of the gears and the angle of the teeth, so that 
stock cutters shall make an exact whole number of teeth. 

" With spur gears it is only necessary to multiply the (circular) pitch of the 
cutter by the number of teeth to be cut to obtain the circumference of the 
gears. With spiral gears this operation gives the length of a portion of a 
spiral, or, more properly, helix, wound upon the pitch surface. We do not 
know the angle of this helix, the diameter of the pitch cylinder upon which 
it is wrapped, or even what part of a complete turn the known portion com- 
prises. The length is known for each gear and nothing more, and it becomes a 
matter of trial to find the diameters of the gears and the helix angle to suit 
this portion of the helix and at the same time to fill the required center distance. 

" Fig. 172 is a conventional representation of the pitch surface of a spiral 
gear, the surface being extended beyond the limits of the gear in order that the 
two helixes with which we are concerned may be shown. The first of these, 
a b c d e f, is the tooth helix and the second, a g h d i p, is the normal helix. 
The tooth helix is of importance because it defines the angle of the teeth. 
Given the diameter of the pitch surface, the helix may be defined by the angle 
k a I or by the length a f, in which it makes a complete turn — that is, by its 
pitch. For the determination of the speed ratio of a pair of gears the former 
method is the more convenient, but the tables supplied with universal milling 
machines which are used in setting up the machine employ the latter method. 

(l In all spiral gear problems we have two pitches to deal with — the pitch of the 



228 



AMERICAN MACHINIST GEAR BOOK 






tooth helix and the pitch of the teeth. The latter may be measured in several 
ways. First is the value a n measured on the circumference or the circular 
pitch, which is analogous to the pitch of spur gears; second is the value a o 
measured on the normal helix or the normal pitch, for which the cutters must 
be selected; third is the value a r measured parallel with the axis or the axial 
pitch. Since the cutters must be selected with reference to the normal pitch, 
the length of the normal helix is naturally of importance in connection with 
the number of teeth in the gear. The normal pitch multiplied by the number 
of teeth must naturally equal the length a g h d of this helix measured between 
its intersections a and d with the helix of a single tooth. Note that the length 
of the normal helix to be considered is the length a g h d between its inter- 
sections with the tooth, and not the length a g h i p q of a complete turn around 
the cylinder. That this is true may be seen by reference to Fig. 173, in which 
the angle k a I is nearly a right angle. It is apparent from this illustration 
that the length of the normal helix from a to d takes in all the teeth, and that 
a 0, multiplied by the number of teeth, must equal a h p d and not a h p q. 
This length a h p d is always less than a h p q, and usually much less. Fig. 
174, A, is a development of Fig. 173 on a reduced scale, a d being the developed 
length of the normal helix. Fig. 174, B, and Fig. i74,C, show how with the same 





fig. 175. 



circumferential pitch and the same number of teeth but a reduced value of the 
angle kal, the length of the normal helix which cuts all the teeth grows shorter 
until it may make but a small part of a complete turn around the cylinder. 
It is clear that in all cases the line a d cuts all the teeth precisely as does the 
circumference a a, which goes completely around the cylinder. It is also 
clear that if the normal pitch is decided upon at the start, a diameter of cylinder 
and a helix angle must be found such that the normal pitch, multiplied by 
the number of teeth, shall equal the length of the normal helix between two 
intersections with the tooth helix. 



SPIRAL GEARS 229 

" It is natural to ask, Why not employ the circumferential pitch and so deal 
directly with the circumference instead of the normal helix? Because we do 
not know what it is. The normal pitch is determined by the cutter used, 
while the circumferential pitch depends also upon the helix angle, and until 
this angle is known the circumferential pitch is not known. 

' ' In the extreme case of a spiral gear in which the helix angle is so small 
that the gear becomes a single thread worm, as in Fig. 175, points and d 
coincide and the length of the helix between a and d becomes the normal 
pitch. It is, however, true as before that the normal pitch, multiplied by the 
number of teeth, which is now one, is still equal to the length of the normal helix 
between two intersections with the tooth helix. 

"A glance at Fig. 174 will show that in gears of the same diameter the length 
of the normal helix * grows shorter as the angle k a I grows less, and hence 
that it and its gear will contain successively fewer and fewer teeth of the same 
normal pitch. That is to say, the number of teeth in a gear varies with the 
helix angle as well as with the diameter, and the number of teeth in two gears of 
the same normal pitch is not necessarily proportional to the diameters. In fact, 
it is never so proportional, except when the angle k a I is equal to 45 degrees. 
The diametral pitch of the cutters and the diameter of the gear thus do not determine 
the number of teeth. 

" The two facts thus developed are fundamental and will bear restating: 

1 ' First, the number of teeth is equal to the length of the normal helix divided by 
the normal pitch. 

" Second, the numbers of teeth in a pair of gears are not proportional to the 
diameters, except when the angle of the tooth helix is 45 degrees. 

THE SPEED RATIO 

" Fig. 176 illustrates the simplest possible case of a pair of spiral gears. The 
gears are of equal size and the tooth helix has an angle of 45 degrees. Such 
a pair of gears will obviously run at the same speed — that is, have a speed 
ratio of 1 — and as obviously both will have the same number of teeth. Now, 
unlike spur gears, there are two ways in which the speed ratio of such a pair 
of spiral gears may be varied. First, the diameters of the gears may be 
changed, as with spur gears, the angle of the tooth helix remaining unchanged, 
as in Fig. 177; and second, the angle of the helix may be changed, the diameters 
of the gears remaining unchanged, as in Fig. 178. These methods act in very 
different ways. The first method is analogous to the procedure with spur 
gears. As with spur gears, the circumferential or pitch-line speed of the two 

♦"Length of normal helix" is to be understood as meaning the length of that helix between 
two intersections with the same tooth helix. 



230 



AMERICAN MACHINIST GEAR BOOK 



gears remains, as before the change, equal, but the length of the circumfer- 
ence of the two gears is unequal and the largest one thus has a less number of 
revolutions than the smaller one. The second method is entirely unlike 
anything seen in connection with spur gears. By it the pitch-line speeds 
of the two gears are made unequal, and hence, while their diameters are 
equal, the lower one revolves the more slowly. This points out another 




Fig. 179 



Fig. 178 



Fig. 177 



THE SPEED RATIO. 



fundamental difference between spiral and spur gears: With spiral gears, 
unless the helix angle is 45 degrees, the pitch-line speeds of two mating gears 
are not the same. 

" The two methods of changing the speed ratio shown in Figs. 177 and 178 
may be combined. That is, part of the desired change in speed may be ob- 
tained by changing the diameters of the gears and the remainder by changing 
the angle of the helix. Given the speed ratio and the diameter of one of the 
gears, we may assume a helix angle and find a diameter for the second gear 
to go with it which shall give the desired speed ratio, and, having done this, 
a second angle may be assumed and a second diameter be found. There are 
thus an indefinite number of combinations of angles and diameters which will 
give the required speed ratio. Note, however, that with the diameter of one 



SPIRAL GEARS 231 

gear fixed, every change in the diameter of the other changes the distance 
between centers, that not every angle of helix can be obtained by the gears 
which are furnished with universal milling machines, and that if ready-made 
cutters are to be used, the lengths of both normal helices must be exact multiple 
of the normal pitch of the teeth. 

" The limitation of the helix angle is not, however, as serious as is usually 
supposed. The tables for spirals which have heretofore been supplied with 
universal milling machines give but a few of the spirals which can be obtained 
with the change gears which are regularly supplied with the machines. For 
universal milling machines, about two thousand spirals can be cut with these 
gears. 

"Geometrically speaking, there is a wide range of choice in the helix angle. 
As regards the desirability of different angles from the standpoint of durability, 
the conditions are essentially the same as in worm gearing. Reference to 
Charts 10 and 11 under worm gears will show that the most favorable angle 
for durability is at about 45 degrees. There is, however, but a trifling increase 
in wear down to 30 degrees, no serious increase down to 20 degrees, and no 
destructive increase down to about 12 degrees. As the angle of worm is the 
complement of the angle of the driving spiral gear, the angle selected from 
Charts 10 and 11, for worm gears, should be the angle of the follower a, which 
is measured from the axis. Where gears are to transmit considerable power 
the best results should attend the use of angles between 30 and 45 degrees, 
while angles as low as 20 degrees may be used without hesitation, and as low 
as 1 2 degrees if the gears are to run in an oil bath or do light work only. The 
angle may also be increased above 45 degrees by similar amounts and with 
similar results. 

"Fig. 179 is a development of the gears of Fig. 178, the angle a of Fig. 179 
being equal to k a I of Fig. 178, but in reversed position, because in Fig. 178 
the upper side of the driver is seen, while in Fig. 179 the direction of the teeth 
is that of the lower side of the driver." 



NOTATION TOR SPIRAL GEARS 

The angle as given for spiral gears is from the axis, which is the opposite 
or complement of the angle for a worm, therefore the angle governing the 
efficiency of spiral gears should be determined from tables on worm gears 
as the angle of the follower (a). 

The greatest angle must always be the driver, except where the angle is 
45 degrees, when either gear may drive. 

All of the tooth parts are derived from the normal pitch. The pitch diam- 



232 AMERICAN MACHINIST GEAR BOOK 

eters are derived from the circular pitch, which is never the same in both 
gears of a pair, except where the angle of both gears is 45 degrees. 

As the diameter of the spiral gear is no indication of its speed ratio, the 
terms gear and pinion are liable to be confusing, therefore follower and driver 
are used. 

N,, = number of teeth in follower. 

N t = number of teeth in driver. 

d 2 = pitch diameter of follower. 

d x = pitch diameter of driver. 

a = angle of follower. 

p = angle of driver. 

p./ = circular pitch of follower. 

p/ = circular pitch of driver. 

p' n = normal circular pitch (the same in both gears of a pair) . 

P = normal diametral pitch (the same in both gears of a pair). 

£ 2 = lead of follower (length of tooth helix). 

L Y = lead of driver (length of tooth helix). 

D 2 = outside diameter of follower. 

D 1 = outside diameter of driver. 

s n = addendum of normal pitch. 

r 2 = revolutions of follower. 

r 1 = revolutions of driver. 

8 = angle of shafts. 

C = center distance. 

EXAMPLES 

Specifications for a pair of spiral gears are sometimes given in this manner: 
Required a pair of spiral gears; ratio 3 to 1, to operate on 5-inch centers. 
The outside diameter of the driven gear must not exceed 7 inches; to be in 
the neighborhood of 6 diametral pitch. 

As the most efficient spiral angle is in the neighborhood of 45 degrees, the 
follower should be made as large as possible, as to obtain this angle the diameter 
of both gears must be in proportion to their number of teeth, as for spur 
gears. As the pitch mentioned in connection with spiral gears is always the 
normal pitch, to obtain a trial pitch diameter for the follower twice the ad- 
dendum of the normal pitch subtracted from the outside diameter wiJl give 
the pitch diameter, according to formula 18: 

d, = D 2 --^=j 1~ = 6^inches. 

and, 

d x = 5 X 2 — 6% = $% inches. 



SPIRAL GEARS 



23.3 







DRIVER 




FOLLOWER 


REMARKS 




TO FIND 


FORMULA 


TO FIND 


FORMULA 




I 





dx Tx 
TaU * = ITVi 


a 


QO° -0 


Axes at right 
angles only. 


2 





P'x 

Tan = . / 
p 2 


a 


90° -0 


Axes at right 
angles onlv. 


3 





p'n 

Cos = -77- 
P 1 


a 


8 -0 




4 





Tan = " r — 
.La 


a 


8 -0 




5 


^ n 


-^r- -cost 


p'n 


d-i ir 
N 2 C0Stt 


Same in both 
gears. 


6 


#'" 


p\ cos 


p'n 


p'l COS O 


Same in both 
gears. 


7 


*'i 


cos 


p'l 


p'n 

cos a 




8 


fi 




P'i 


d-i ir 

Ni 




9 
10 


Li 

Li 


p'i Nx 
d\ it tan a 


U 
Li 


p'x Ni 

di ir tan 


Axes at right 
angles only. 


11 


#1 


di P cos 


N* 


di P cos a 




12 


iVi 


d\ ir 

~7T 


Ni 


di ir 
p'i 






A 


2 c 


di 


2C 






/'1 \ 

1 — tow a ) + 1 




13 


{y 2 t™ * Y x 


Axes at right 
angles only. 




A 


2C 


di 


2 C — di 




14 


\ r 2 cos a J 




15 


rfi 


Ni p\ 0.3183 


di 


Ni p'i 0.3183 






dx 


Nx 


di 


Ni 




16 


P cos 


P COS a 




17 


Dt 


dx + 2S n 


Di 


di + 2 s n 




18 


- JDi 


2 

dx + -p 


D* 


2 
di + -p 


14^° standard 
only. 


19 


Cutter 
Seecharti4 


Nx 

COS 3 

Nx 


Cutter 

Ni 
2 P cos a 


Ni 
cos 3 a 




20 


^ ' 2 P COS ~ T " 





Formulas for Spiral Gears. 



234 AMERICAN MACHINIST GEAR BOOK 

The next step is to find the angle of driver by formula i. 

Tan P = HX = 6HXV = J ' 5 ' ° r s6 I9 • 
The angle of follower =90°— 56 i9 /= 33° 4 1 '- 
Find the provisional number of teeth by formula n. 

N x = d x P cos f3 = $14 X 6 X 0.5546 = 11.092. 
A r 2 = d. 2 P cos a = 6% X 6 X 0.8321 = 33.284. 

Naturally the number of teeth must be whole numbers, so it will be necessary 
to change either the center distance, or to make numerous calculations and 
shift the diameters. Practically, however, it is possible to have quite an error 
in the normal pitch; the normal pitch, or the pitch of the cutter, preferably 
being under size rather than over. The teeth are thus cut enough deeper than 
standard to secure the proper thickness of tooth at the pitch line. 

This difference may be 0.02 of the circular pitch in some cases. 

If the cutter is heavier than the normal pitch it will be impossible to secure 
enough clearance at the bottom of the tooth as the proper thickness of tooth 
will be reached before getting the depth of tooth required. 

If the center distance can be changed the pitch diameters may be shifted 
by the method explained on page 66 of Mr. F. A. Halsey's book — " Worm and 
Spiral Gearing," as follows: 

final diameter n 

provisional diameter 11.094 
or 

final diameter = provisional diameter X 

That is: 

' final d x = iVz X ■ = 3.305; 

11.094 o jy 

and, 

final d = 6V 3 X 33 Q = 6.610; 
33.282 

and 

fa -j- d 2 = 3.305 -f 6.610 = 9.915 = twice the corrected center distance. 

In the present example the normal circular pitch for the nearest even number 
of teeth, n and ^, by formula 5 would be: 

d t tt ?M X 3.1416 vx , 

p- = _jl_ cos p = ^ J + x 0.5546 = 0.5280. 

As the pitch of the cutter is 0.5236, this error will not prevent a first-class job 
being turned out if proper precautions are taken, and no change will be re- 
quired in the center distance. 



11 



11.094 



SPIRAL GEARS 



235 



These points being settled, the remaining calculations are simple. Before 
making any calculations, the requirements should be put in the form of a table 
to avoid confusion, as follows: 



DIMENSIONS 



Pitch diameters 

Revolutions 

Angles 

Number of teeth .... 

Circular pitch 

Normal pitch 

Cutter used 

Lead, exact 

Lead, approximate. . . 

Addendum 

Outside diameter .... 
Whole depth of tooth 
Thickness of tooth . . . 

Gear on worm 

First gear on stud . . . 
Second gear on stud . 
Gear on screw 



DRIVER 


FOLLOWER 


3.K 


6K 


3 
56° i 9 ' 


33°\i' 


ii 
0.9520" 
0.5280" 


33 
0.6345" 
0.5280" 


No.2-6/> 

6.9795" 
6.9670" 
0.1680" 


No.3-6/> 

31.4160" 

31.5000" 

0.1680" 


3.6690" 
0.3630" 
0.2640" 


7.0030" 
0.3630" 
0.2640" 


86 


72 


48 
28 


40 
56 


72 


3 2 



An error 'of 0.5 inch in a lead of 50 inches would not ordinarily be pro- 
hibitive, but the angle must be changed to suit any alteration of the lead 
or the cutter will drag. If too much alteration is made in the lead and angle, 
the teeth must be cut a little deeper than standard to allow the gears to assem- 
ble on the proper shaft angles. 

The amount of adjustment that can be made depends, of course, upon the 
accuracy required, and should be done by some one accustomed to the work. 
This is not possible when cutting helical or herringbone gears, as the tooth has 
contact the entire length of face and a slight error is noticeable. The accuracy 
of the final calculations may be checked by the angles, obtained from the 
circular pitch by formula 3. 

Another way of presenting this problem is as follows: 

Required, a pair of spiral gears; ratio 4 to 1; about 8 diametral pitch 
(0.3927-inch circular pitch). Angle of spiral for driver, /?, to be about 55 
degrees, a = 90 — /3 = 35 degrees. 

Find the diameter of driver by formula 13. 



2C 2 X 6 

I— - tan a J + 1 j — X 0.7002 J + 1 



= 3.1572 inches. 



236 AMERICAN MACHINIST GEAR BOOK 

The diameter of follower d 2 = 2 C — d % = 12 — 3.1572 = 8.8428 inches. 
The remaining dimensions are found as in the first example. 

Still another example: 

The ratio of a spiral gear drive is 4 to 1. The diameter of the driver 
cannot be less than 8 inches, on account of the size of the shaft. The distance 
between centers to be 53^ inches. No pitch mentioned. 

Assumed diameter of driver d x = 8 inches. 

Diameter of follower rf s = (5^ X 2) - 8 = 3 inches. 

According to formula 1 : 

Tan B = -^ = 8X4 = 10.66 or 86° 25'. 
d,r, 3X1 

Try 7 diametral pitch: 
According to formula 1 1 : 

N x = d x P cos P = 8 X 10 X 0.0625 = 5 teeth; 
iV 2 = d 2 P cos a = 2 X 10 X 0.9980 = 19.96, say 20 teeth; 
which just happens to come out even. 

If the center distance is not specified, the best plan is to assume number of 
teeth, angles, and pitch of cutter, P or p' n and find the corresponding center 
distance by formula 20. 

Example : 

What center distance will be required for a pair of spiral gears 11 and $$ 
teeth, 6 diametral pitch, the angle of the 11 tooth drive being 56 19' and the 
angle of the follower 33 41 r . 

According to formula 20: 

r = N i . N * = " , S_3 = 6 

" 2 P cos P "*" 2 P cos a " 2 X 6 X 0.5546 T 2X6X 0.8321 " * 5 9 
+ 3.3049 = 4.9578 inches center distance; 

1.6529 being the pitch radius of the pinion, and 3.3049 the pitch radius of the 
gear. 

When the center distance is approximate, this is the simplest solution of 
the problem, the speed ratio being used in place of a trial number of teeth, and 
the number of teeth made to suit the desired center distance. 

A CHART FOR LAYING OUT SPIRAL GEARS 

Chart 13 with the following explanation of its deviation and use will be an 
aid in solving spiral gear problems once the provisional numbers of teeth are 
obtained. This diagram and explanation were originally published in Amer- 
ican Machinist, February 27, 1902, by J. N. Le Conte. 

" The provisional numbers of teeth will not in general be whole numbers, 



SPIRAL GEARS 



237 



Ni 



2 PC 




.5 .6 I .7 

60 cos a 5Q 
Follower 



CHART 13. DIAGRAM FOR LAYING OUT SPIRAL GEARS. 



1.0 
10°0° 

2 PC 



but we must choose the nearest whole numbers to the ones obtained, and 
recalculate the angles and radii to fit the new case, as has been previously 
shown. The direct solution of this depends upon the solution of the equation: 



= 2 PC 



COS a 



COS (8 — a) 

" In which N 1 and TV", are the nearest whole numbers of teeth to the calculated 
ones, and 8 is the shaft angle, or 8 = a + ft. As is well known, this equation 
cannot be solved by any simple means, for it is of the fourth degree, and, 
though possible of solution, such solution is not practical. Furthermore, 



238 AMERICAN MACHINIST GEAR BOOK 

there are four real values of a which will satisfy it. Graphic methods of solu- 
tion, or continued approximations, must then be resorted to. Chart 13 
gives a method by which the angle can be read off directly. Having obtained 
the nearest whole number of teeth on the gears, find on the diagram the point 

N N 

G whose co-ordinates are — ——=, and — =^, on the inner scales. Through this 

point draw a line or merely lay a straight-edge tangent to the curve repre- 
senting the shaft angle. The outer scales on the bottom and left will give 
roughly the angles a and fi respectively, and the inner scales the values of 
cos a and cos ft quite accurately. The radial lines of velocity ratio will 
facilitate the location of the desired point, for if the ratio be one of those given, 
the point must lie on its line. It will also be noticed that for shafts crossing 
at 90 the position of the line gives the desired angles at its two extremities. 

"It is interesting to note that two lines can be drawn through a given point 
tangent to the curves, as shown. As a matter of fact, four such lines could 
be drawn provided the whole of the curves were laid in, but that portion shown 
is the only portion giving positive angles, i. e., angles within the angle 8. 
But there will be two separative positive values of the angle a, which, with 
a given velocity ratio, number of teeth and shaft distance, will work cor- 
rectly together, giving of course different values of the radii. Which of these 
is the one required can always be told as lying nearest to the first approxima- 
tion of the angle. If the point G lies on one of the curves, the two positions 
coincide (a limiting case), and if it lies on the concave side the solution is 
impossible within the angle 8. 

" As an example of the use of the diagram, take the oft-quoted case of Mr. De 
Leeuw. Here angular velocity 

y 1 = — = H, N, = 8, N, = 32, C = 4.468", 8 = 90 , and P = 6. 
7i 
Then 

N, N , 

Ypc = °' 149 ' Ypc = °' 596 - 

" These co-ordinates give the point G on the diagram. A line through G 
drawn tangent to the lower part of the 90 curve gives quite accurately cos 
a = 0.894, and cos ft = 0.447, or: 

a = 26 35' and ft = 63 25', 
agreeing quite closely with the result derived analytically. If the second line 
be drawn through G tangent to the upper portion of the curve, it gives: cos a 
= 0.787 and cos ft = 0.617, or: 

a = 3 8° 61' and fi =-- 51 54'. 
These fulfill the requirements." 



SPIRAL GEARS 



239 



SPIRAL GEAR TABLE 

While it is better in every case to understand the principles involved before 
using a table, as this tends to prevent errors, they can be used with good results 
by simply following the directions carefully. The subject of spiral gears is 







To obtain the 




To obtain the 








To obtain the 


pitch diame- 


To obtain the lead of spiral, 


pitch diame- 








circular pitch 
for one tooth 


ter, divide by 
the required 
diametral 


divide by the required dia- 


ter, divide by 
the required 
diametral 


To obtain the 
circular pitch 






divide by the 


pitch and 


metral pitch and multiply 


pitch and 


for one tooth 






required d i - 
a m e t r a ] 


multiply the 




multiply the 


divide by the 






quotient by 
the required 


the quotient by required 


quotient by 
the required 


required dia- 






pitch. 


number of 
teeth. 


number of teeth. 


number of 
teeth. 


metral pitch. 




ANGLE OF 


CIRCULAR 


ONE TOOTH OR 




ONE TOOTH OR 


CTRCULAR 


ANGLE OF 


SPIRAL 






LEAD OF SPIRALS 






SPIRAL 


DEGREES 


PITCH 


ADDENDUM 




ADDENDUM 


PITCH 


DEGREES 


Small 


Small 


Small 


Small 


Large 


Large 


Large 


Large 


Wheel. 


Wheel. 


Wheel. 


Wheel. 


Wheel. 


Wheel. 


Wheel. 


Wheel. 


1 


3-1419 


I.OOOI 


180.05 


3.1420 


57.298 


180.01 


89 


2 


3-1435 


1.0006 


90.020 


3-1435 


28.653 


90.016 


88 


3 


3-1457 


1. 00 1 3 


60.032 


3-I458 


19.107 


60.026 


87 


4 


3-I49I 


1.0024 


45-038 


3.1492 


14-335 


45-035 


86 


5 


3-1535 


1.0038 


37-077 


3-I527 


H-473 


36.044 


85 


6 


3.1.I89 


i-oo.S5 


30.056 


3-I589 


9.5667 


30.055 


84 


7 


3-1652 


1.0075 


25.728 


3-1651 


8.2055 


25-778 


83 


8 


3-I724 


1.0098 


22.573 


3-I724 


7.1852 


22-573 


82 


9 


3.1806 


1.0124 


20.082 


3-1807 


6.3924 


20.082 


81 


10 


3. 1 goo 


1.0154 


18.092 


3. 1 901 


5-7587 


18.092 


80 


II 


3.2003 


1.0187 


16.464 


3-2003 


5-2408 


16.464 


79 


12 


3-2145 


1.0232 


15.076 


3-2105 


4.8097 


15.104 


78 


13 


3.2242 


1.0263 


13.966 


3.2294 


4-4454 


13.988 


77 


14 


3-2377 


1 .0306 


12.986 


3-2378 


4-1335 


12.986 


76 


15 


3-2522 


1-0352 


12.138 


3-2524 


3-8637 


12.138 


75 


16 


3.2679 


1.0402 


n-393 


3-2678 


3.6279 


H-397 


74 


17 


3.2848 


1.0456 


10.417 


3.2821 


3-4203 


IO-745 


73 


18 


3-3ii6 


1-0514 


10.192 


3-3032 


3-2360 


10.166 


72 


19 


3-3225 


1.0576 


9.6494 


3-3225 


3-0715 


9.6494 


7i 


20 


3 -343° 


1. 0641 


Q.I 848 


3-3433 


2.9238 


9.1854 


7o 


21 


3-3650 


1.0711 


8.7662 


3-3652 


2.7904 


8.7663 


69 


22 


3-3882 


1.0785 


8.3862 


3-3833 


2.6694 


8.3862 


68 


23 


' 3-4127 


1.0863 


8.O399 


3.4129 


2-5593 


8.0403 


67 


24 


3-445.1 


1 .0946 


7-7379 


3-4391 


2.4585 


7.7242 


66 


25 


3-466l 


1.1033 


7-4332 


3-4663 


2.3662 


7-4336 


65 


26 


3-4953 


1.1126 


7.1664 


3-4952 


2.2811 


7.1663 


64 


27 


3-5258 


1. 1223 


6.9198 


3-5257 


2.2026 


6.9197 


63 


28 


3-5579 


1-1325 


6.6912 


3-5575 


2.1300 


6.6916 


62 


29 


3-5918 


I-I433 


64709 


3-5919 


2.0626 


6.4799 


61 


30 


3.6276 


I-I547 


6.2778 


3.6277 


2.0000 


6.2832 


60 


31 


3.6650 


1. 1666 


6.0979 


3-6652 


1. 9416 


6.0997 


59 


32 


3-7043 


1.1791 


5.9282 


3-7044 


1.8870 


5.9282 


58 


33 


3-7457 


1. 1923 


5-77IO 


3-7459 


1.8360 


5-768o 


57 


34 


3-7894 


1.2062 


5.6181 


3-7826 


1.7882 


5-6178 


56 


35 


3-8349 


1.2207 


5-4754 


3-835I 


1-7434 


5-4770 


55 


36 


3.8830 


1.2360 


5-3431 


3-8834 


1. 7013 


5.3448 


54 


37 


3-9336 


1. 2521 


5.2201 


3.9261 


1.6616 


5.2200 


53 


38 


3.9867 


1.2690 


5.1028 


3.9921 


1.6242 


5.1026 


52 


39 


4.0482 


1.2867 


4.9866 


4.0416 


1.5890 


4.9920 


5i 


40 


4.1010 


I-3054 


4.8873 


4.1012 


1-5557 


4.8874 


50 


41 ! 


4.1626 


1-3250 


4-7885 


4.1540 


1.5242 


4.7884 


49 


42 


4.2273 


1-3456 


4.6949 


4.2272 


1.4944 


4.6948 


48 


43 


4.2956 


1-3673 


4.6065 


4.2956 


1.4662 


4.6062 


47 


44 


4-3671 


I-390I 


4-5223 


4-3675 


t-4395 


4-5225 


46 


45 


4.4428 


1.4142 


4.4428 


4.4428 


1.4142 


4.4428 


45 



Table 27 — Spiral Gear Table Shaft Angles 90 Degrees 

For one Diametral Pitch. 



240 AMERICAN MACHINIST GEAR BOOK 

so much more complicated than other gears that many will prefer to depend 
entirely on tables. 

This table gives the circular pitch and addendum or diametral pitch and 
lead of spirals for one diametral pitch and with teeth having angles from i 
to 89 degrees to 45 and 45 degrees. For other pitches divide the addendum 
given and the spiral number by the required pitch, and multiply the results 
by the required number of teeth. This will give the pitch diameter and lead 
of spiral for each gear. For the outside diameter add twice the addendum of 
the normal pitch, as in spur gearing. 

Suppose we want a pair of spiral gears with 10 and 80 degree angles, 8 
diametral pitch cutter, with 16 teeth in the small gear, having 10-degree angle 
and 10 teeth in the large gear with its 80-degree angle. 

Find the 10-degree angle of spiral and in the third column find 1.0154. 
Divide by pitch, 8, which is 0.1269. Multiply this by the number of teeth; 
0.1269 X 16 = 2.030 = pitch diameter. Add two addendums or f = 0.25 
inch. Outside diameter = 2.030 + 0.25 = 2.28 inches. 

The lead of spiral for ten degrees, for small gear, is 18.092. Divide by 

pitch = — ^— = 2.2615. Multiply by number of teeth, 2.2615 X 16 = 

o 

36.18, or lead of spiral, which means that the tooth helix makes one turn in 
36.18 inches. 

For the other gear with its 80-degree angle, find the addendum, 5.7587. 
Divide by pitch, 8 = 0.7198. Multiply by number of teeth, 10 = 7.198. 
Add two addendums, or 0.25, gives 7.448 as outside diameter. 

The lead of spiral is 3.1901. Dividing by pitch, 8, = 0.3988. Multiply 
by number of teeth = 3.988 the lead of spiral. 

When racks are to mesh with spiral gears, divide the number in the circular 
pitch columns for the given angle by the required diametral pitch to find the 
corresponding circular pitch. 

If a rack is required to mesh with 40-degree spiral gear of 8 pitch, look 
for circular pitch opposite 40 and find 4.101. Dividing by 8 gives 0.512 as the 
circular pitch for this angle. The greater the angle the greater the circular 
or linear pitch, as can be seen by trying an 80-degree angle. Here the cir- 
cular pitch is 2.261 inches. 



SPIRAL GEARS 



241 





CORRESPONDING 


CORRESPONDING 




PITCH OF CUTTER 






ADDENDUM 




CIRCULAR PITCH 


DIAMETRAL PITCH 




P 


P' 


P 


S n 


2 


2.2214 


I.4142 


.70710 


2K 


1-9745 


I.5909 


.62853 


2V 2 


1.7771 


I.7677 


.56568 


2% 


1.6156 


1-9445 


.51426 


3 W 


1.4809 


2.1213 


.47140 


3^ 


1.2694 


2.4748 


.40406 


4 


1.1107 


2.8284 


•35355 


5 


.8885 


3-5355 


.28284 


6 


.7404 


4.2426 


•23570 


7 


•6347 


4-9497 


.20203 


8 


•5553 


5.6568 


.17677 


9 


•4936 


6.3638 


•I57I3 


10 


•4443 


7.0710 


.14142 


12 


.3702 


8.4853 


.II785 


14 


•3173 


9.8994 


.IOIOI 


16 


.2776 


ii-3i37 


08838 


18 


.2468 


12.7279 


.07856 


20 


.2221 


14.1421 


.07071 


22 


.2019 


I5-5563 


.06428 


24 


.1851 


16.9705 


.05892 


26 


.1708 


18.3847 


•05439 


28 


.1586 


19.7990 


.05050 


30 


.1481 


21.2116 


.04714 


32 


.1388 


22.6274 


.04419 


36 


.1234 


254558 


.03928 


40 


.1111 


28.2842 


•03535 


48 


.0925 


33-94H 


.02988 



Table 28 — Spiral Gears of 45 Degrees . 
For determining the pitch diameters of spiral gears when the pitch of cutter is assumed and 
angle of spiral is 45 degrees: Multiply the addendum of the normal pitch found in fourth column 
by number of teeth. 

DIRECTION OF ROTATION AND THRUST OF SPIRAL GEARS 

The use of spiral gears generally causes some study on the part of the de- 
signer to determine the proper direction of the teeth, having given the direction 
of rotation of the two shafts which are to be connected. Another point about 
spiral gears which also causes some study after the direction of the teeth has 
been determined is the direction of the axial thrust, that is, the direction 
in which the spiral gears tend to move along their axes when transmitting 
motion. The proper direction of thrusts is very important to locate correctly 
the ball-thrust bearings of other suitable anti-friction devices. 

It sometimes occurs that a newly designed machine, when started for the 
first time, has a shaft which is driven by spiral gears running in the opposite 
direction to that which was intended, or that the anti-friction washers have 



242 



AMERICAN MACHINIST GEAR BOOK 



been located on the wrong side of the helical gear. To obviate this and 
reduce the chance for mistakes in directions of rotation and strains in spiral 
gears, four diagrams, 180 to 183, are arranged to readily illustrate every pos- 




g Driver 



FIG. 180. 



FIG. 181. 



RIGHT-HAND SPIRAL GEARS. 



sible combination; giving the direction of the teeth, their rotation, and the 
direction of the lateral strains when they are transmitting motion in the 
directions indicated. These diagrams ehminate the necessity of consulting 




Driver 



FIG. 182. 



r\ ' J 



FIG. 183. 



LEFT-HAND SPIRAL GEARS. 



a gear model, nor is it necessary to go through a series of hand manipulations 
describing the rotations in the air. 

In the diagrams, Figs. 180 and 181 represent a pair of right-hand helical 
gears with the direction of rotation of the drivers reversed. 



SPIRAL GEARS 



243 





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3 v 












So 








*/» 


\ 






s 






V * 












5 










v9a 
















^,- S 












^ 
















\ 






s 




r ^ 












3 










w 


\ 




\ 






s 




3 


\ 










v- 










\ 


\ 


























5 











\ 


-\ 












\ 






r 1 1 Hill 1 1 




































y 1- - 


II ifiniii ||i in in 










°S 


















\ 






V--V- 




1ml 1 










tH 




















s 




_S____^_ 


N T TV 


llllllllllfHUIIIIIIIIIll 











«o t« x ffl oh w eo-#»o 

T*i *" ' T^ 3""^ ^"^ ,— ' 



- §5 w> ^ -* S eo 



Number of Teeth in the Spiral Gear 



O O O O C3 C3 
t-. ^00 OS O JH CSj 



The diagrams Figs. 182 and 183 each show a pair of left-hand helical 
gears, also with the directions of their drivers reversed. It should be noted 
that reversing the direction of rotation of the drivers reverses the directions 
of their axial thrusts. Also, if the driven gears are made the drivers and 
rotating in the same direction, as shown, the lateral strains are also reversed; 



244 AMERICAN MACHIXI5T GEAR BOOK 

that is, if in Fig. 1S0 the driven gear is made the driver and rotates as 
indicated, the gear marked the driver, which is now the driven gear, will 
rotate as shown, but the axial thrust of each gear would be as in Fig. 
i Si. If the driven gear of Fig. 1S1 is made the driver, the lateral strains 
are as shown in Fig. 1S0. This is also true of the left-hand combinations 
shown in Figs. 182 and 183. originally published in the American Machinist 
by William F. Zimmerman. 



SECTION IX 



Skew Bevel Gears 



Bevel gears which do not have their axes in the same plane, popularly 
known as skew bevel gears, present one of the most complicated construc- 
tions in gearing. Ordinarily, this arrangement consists of a common bevel 
pinion meshing with a bevel gear having teeth of a spiral type (see Fig. 
184), the contact taking place on a plane parallel to that of, but somewhat 
removed from, the one in which lies the axis of the skew bevel gear. 




FIG. 184. SKEW BEVEL GEARS. 

The contact action of this combination is somewhat better than that in 
spiral gearing, as the sliding action of the latter is replaced by a combined 
rolling and sliding action. The pitch surfaces of the bevel pinion are frusta 
of a figure generated by the revolution of a straight line about an axis to 
which it is not parallel, a "hyperboloid of revolution." 

A typical plan view of a skew bevel gear with its pinion in position is shown 

245 



246 



AMERICAN MACHINIST GEAR BOOK 



as Fig. 185, in which the dimension A represents the offset of the pinion. The 
apex point of the pinion lies in the perpendicular axis plane of the gear, to 
which point also converge the profile planes of the gear teeth actually in 
mesh with the pinion. It is obvious then that the profile planes of each 
succeeding tooth, when in mesh, must converge to the same point. This 
results in a circle of apexes for the gear having a radius equal to the offset 







FIG. 18 ^. DIAGRAM OF SKEW BEVEL GEARS. 



of the pinion — i.e., the succeeding converging tooth profiles, if prolonged, 
would all be tangent to a circle having a diameter equal to twice the offset 
of the pinion. 

The pinion differs in no way from a regular bevel gear and. therefore, 
governs the proportions of the skew bevel gear. If the pinion was not offset, 
the combination simply a set of bevel gears, the pitch diameter of the gear 
would be F-G. The number of teeth in the skew bevel gear is therefore the 
same as would be required for an ordinary bevel gear of such pitch diameter. 
The actual pitch diameter of the skew bevel gear is considerably greater than 
this "equivalent pitch diameter," depending upon the amount which the 
pinion is offset. 

The normal pitch of the gear (B-C) must conform to the circular pitch of 



SKEW BEVEL GEARS 247 

the pinion, but the circular pitch (D-E) of the gear depends upon its actual 
pitch diameter, the number of teeth being fixed by the pinion and equal to 
the number of teeth required for a common bevel gear of a pitch diameter 
equal to F-G. 

The sliding action of the teeth upon one another also depends upon the 
amount of offset to the pinion shaft. In the combination illustrated in Fig. 
185 it is evident that sliding of the extreme end of the pinion teeth will occur 
from I to D on the gear. This sliding action is accentuated with any increase 
in the dimension A, the limiting condition being when A equals the radius 
of the gear, in which case there would be sliding contact only and no turning 
moment. On the other hand, when A equals zero the sliding action is elimi- 
nated and there is only true rolling contact. 

The principal relationships of the various dimensions, angles, etc., of skew 
bevel gears follow: 

NOTATIONS FOR SKEW BEVEL GEARS 

p = diametral pitch. 

n = number of teeth in pinion. 
N = number of teeth in gear. 
A = offset of pinion shaft. 

6 = angle of offset. 
d' = pitch diameter of pinion. 
D' e = equivalent pitch diameter of gear. 
D' = pitch diameter of gear. 

d = outside diameter of pinion. 
D = outside diameter of gear. 
p' = circular pitch of pinion. 
pi n = normal pitch of gear. 
p\ = circular pitch of gear. 
Ei = center angle of pinion. 
Fi = face angle of pinion. 
C\ = cutting angle of pinion. 
£ 2 = center angle of gear. 
F 2 = face angle of gear. 
C 2 = cutting angle of gear. 
E\ = contact angle of gear and pinion = Ei. 

J = angle increment. 
K = angle decrement. 
Vi — diameter increment of pinion. 

s = addendum. 



248 AMERICAN MACHINIST GEAR BOOK 

FORMULAS FOR SKEW BEVEL GEARS 

(1) 
(2) 
(3) 
(4) 
(5) 

(6) 
(7) 

(8) 

(9) 
(10) 

(11) 
(12) 
D = D' + 2F1, or for greater accuracy = D r -\ n , e - (13) 

£2 = (90 - £1). (14) 

C 2 = (90 - EO - K. (15) 

[(90 - £1) + J)D' , ,. 

£ 2 = ^ • (16) 

DISCUSSION OF FORMULAS 

The formulas for ascertaining the various dimensions and angles of the 
pinion are similar to those for any bevel gear, once the center angle is 
obtained. 

The "equivalent pitch diameter" of the gear is the same as the pitch 
diameter of the regular bevel gear that would give the required speed ratio, 
and is obtained by dividing the number of teeth in the gear by the diametral 
pitch. 

The angle of offset is the angle between the axis plane of the gear and a 
plane passing through the axis of the gear and the common contact point of 



d 


= 


n 
J 


D' e 


= 


N 
P' 


Tan. 6 


= 


2A 

D' e ' 


D' 


= 


2A 

sin. 6 


P\ 


= 


3.1416 iy 

N * 


Tan. Ei 


= 


d f 


1 'in. J 


= 


2 sin. Ei 
n 


Tan. K 


= 


2.314 sin. Ei 


n 


Fi 


= 


Ei + J. 


d 


^= 


Ei - K. 

d' 


s 




n 


V 1 


= 


s cos. Ei. 



SKEW BEVEL GEARS 249 

the pitch diameters (outer) of the opinion and gear. Its tangent is obtained 
by dividing the offset of the pinion shaft by half the equivalent pitch di- 
ameter of the gear, or twice the offset of the pinion shaft divided by the 
equivalent pitch diameter of the gear. 

The pitch diameter of the gear is then obtained by dividing twice the pinion 
shaft offset by the sine of the angle of offset. 

The circular pitch of the gear is equal to the quotient of the pitch circum- 
ference by the number of teeth. 

The tangent of the center angle of the pinion is found by dividing the pitch 
diameter of the pinion by the equivalent pitch diameter of the gear. 

The outside diameter of the gear is usually found by adding to its pitch 
diameter twice the diameter increment of the pinion. This method is not 
quite accurate, as the diameter increment of the gear and pinion is only the 
same when there is no offset to the pinion shaft. The greater this offset is, 
the proportionally smaller does the diameter increment of the gear become. 
A more accurate way of ascertaining the outside diameter of the gear is by 
the use of the second formula. This more accurate method is not absolutely 
correct, however, for it is based on the assumption that the decrease in di- 
ameter increment is proportional to the ratio of the equivalent pitch diameter 
to the pitch diameter of the gear, which is not absolutely true. The possible 
error is so small for any standard gear, however, as to be quite immaterial. 

The center angle of each individual gear tooth is equal to the complement 
of the center angle of the pinion, so that the center angle of a skew bevel 
gear may be taken as the complement of the center angle of the pinion with 
which it is to run. 

The cutting angle of the skew bevel gear is likewise the same for each 
individual tooth and is equal to the difference between the center angle of the 
gear and the angle decrement. 

The face angle of the skew-bevel gear as obtained by the formula given 
is not absolutely accurate, but the error is sufficiently trivial to be safely 
overlooked in practice, unless the face of the pinion is unusually wide and 
the pitch equally small. In such a case, the cut-and-try method of fitting 
the gear to the pinion is advisable, as the calculations involved for an 
accurate mathematical solution are extremely complex. 

The face angle of a skew bevel gear would not be the same as that of a 
bevel gear matched to mate with the skew gear pinion unless the offset of 
the pinion shaft were zero. Such a condition, which would be that existing 
between a set of common bevels of proper proportions, would fix the minimum 
face angle for a skew bevel gear. The maximum face angle would occur with 
the pinion shaft's offset equal to half the pitch diameter of the gear and would 



250 AMERICAN MACHINIST GEAR BOOK 

be one of 90 degrees. Between these limits the face angle of a skew bevel 
gear may be anything, depending upon the difference in the pitch and equiva- 
lent pitch diameters of the gear. Formula (16) is derived on the assump- 
tion that the increase in face angle of the skew bevel gear, from that of a set 
of bevel gears of similar pitch diameters to the condition where there would 
be no rolling action, is governed by the ratio of the pitch diameter of the gear 
to its equivalent pitch diameter. This relationship is only approximately 
accurate, for the actual increase in face angle is not quite constant between 
its minimum and maximum values. For all practical shop requirements, 
however, formula (16) may be considered as correct. Any possible error 
that might arise would but slightly affect the total depth of tooth at the 
small end of the gear where it would be least noticeable or harmful. 

EXAMPLE IN THE DESIGN OF SKEW BEVEL GEARS 

Required a pair of skew bevel gears; 10 diametral pitch, 85 teeth in gear, 
13 teeth in pinion; pinion shaft offset ij^ inches. 

FORMULA 

13 
Pitch diameter of pinion, d f = — = 1.3 . (1) 

Equivalent pitch diameter of gear, D' e — — = 8.5". (2) 

2X1.^ 

Angle of offset, 6, Tan. d = — 5 — — = 0.3529 (3) 

B = 19 26'. 

2X1.^ 

Pitch diameter of gear, D' = — = 9.01 ", say 9.0". (4) 

Circular pitch of gear, p\ = — — tt- = 0.33". (5) 

1.3 

Center angle of pinion, Ei, Tan. Ei = — • (6) 

8-5 

= 0.1529 

E 1 =. 8° 42'. 

_ _ _ 2X 0.15126 , . 

Angle increment, /, Tan. J = (7) 

= 0.02327 
7 = i° 2o r . 

1 -, Tr r~ Tr 2 '3 T 4 X O. I 5 1 2 6 

Angle decrement, A, Tan. K = (8) 

= 0.02692 
K = i° 33'- 



SKEW BEVEL GEARS 251 

Face angle of pinion, F\ = (8° 42') + (i° 20') = io° 2'. (9) 

Cutting angle of pinion, d = (8° 42') - (i° 33') = f 9'. (10) 

Addendum, s = — = 0.1". (11) 

Diameter increment of pinion, V\ = 0.1 X 0.1513 = 0.01513". (12) 

Outside diameter of gear, D = 9 + 2 X 0.015 13 ] 

2 X 0.01513 X 8. 5 \ = 9.03". (13) 

or, iy — 9 t 

Center angle of gear, E 2 = (90 — 8° 42') = 8i° 18'. (14) 

Cutting angle of gear, C 2 = (90 - 8° 42') - i° 33' = 79 45'. (15) 

v 1 f 7? [(90 ~ 8° 42') + i° 2Q'] 9 .0 , , ,s 

Face angle of gear, t 2 = —5 = 87 29 . (16) 

MACHINING SKEW BEVEL GEARS 

Any of the machines used for cutting the ordinary type of bevel gear can 
be used for machining skew bevel gears, if simple adjustments or modifica- 
tions are made. The carrying spindle of the machine must be offset from the 
plane of the cutting tool by a distance equal to the offset of the pinion shaft 
The subsequent operations of cutting the teeth are similar to those employed 
in cutting plain bevel gears, the rotary adjustment of the gear being governed 
by the circular pitch of the gear, not its normal pitch which corresponds to 
the circular pitch of the pinion. The adjustments are somewhat more com- 
plicated than when cutting the simpler gears and must be performed with 
great care, as there is no common apex toward which to work. This adds to 
the difficulties of accurate workmanship and is the main reason why skew 
bevel gears are so seldom employed. 

Another method of designing and cutting skew bevel gears that is some- 
times employed is to make both the teeth of the gear and the pinion of 
spiral type. When this is done, the degree of obliquity of the teeth in the 
gear and in the pinion is made the same in order to facilitate manufacture 
and design. A layout for such gears is shown in Fig. 186. 

These gears are turned up according to the dimensions for bevel gears of 
the same number of teeth, pitch and ratio, and no alteration in the diameters 
is usually made or is any alteration in the angles necessary, due to the fact 
that though the apex points of the two gears do not coincide, the converging 
conical surfaces are parallel to those of bevel gears with a common apex 
point. 

Both gear and pinion are machined with the plane of the cutting tool offset 
from the carrying spindle of the machine. This offset is different for the 



252 



AMERICAN MACHINIST GEAR BOOK 



two gears if their speed ratio is other than i to i . For gears of similar dimen- 
sions, the total offset of the shafts would be divided in two and the correct 
offset between the spindle of the machine and the cutting tool plane would 
be one-half the total offset for both gears. For any other speed ratio, the 
total offset is divided proportionally to the ratio, the smaller offset being 
employed for cutting the pinion and the larger for machining the gear. For 
instance, when cutting skew bevel gears having a • shaft offset of 2 inches 




American Ifachinirt 



FIG. 186. LAYOUT FOR SKEW BEVEL GEARS. 



and a speed ratio of 2 to 1, the machine drop or offset for the pinion 



would be 



2 + 1 



= 0.666 inch and for the gear, 0.666 X 2 = 1.333 inches. 



Skew bevel gears cut according to this apportioning method have proved 
very satisfactory, and the only criticism that can be advanced is on account 
of the decreased strength of the teeth as the gears are usually cut. The 
teeth being inclined to the circumference of the gear — that is, not being radial 
— 'the- circular pitch must necessarily be greater than that of common bevel 
gears of similar proportions, for the circular pitch of the common bevels 
corresponds to the normal pitch of the skew bevel gears. This would neces- 
sitate an increase in diameters, the amount of increase depending upon the 
angularity of the teeth. If this is attended to, the full strength of the teeth 
will be developed. 



SKEW BEVEL GEARS 253 

The obliquity of the teeth of skew bevel gears of all varieties is the cause 
of one other annoyance, due to the unavoidable sliding action between the 
teeth. It has been found that if the common 143^-degree involute tooth is 
used the teeth do not clear properly. This has been overcome by making the 
angle 20 degrees. In extreme cases an even greater angle might have to be 
employed, but for any ordinary installation of skew bevel gears, the adoption 
of the 20-degree involute tooth will allow the teeth to clear satisfactorily. 



SECTION X 

Intermittent Gears 

Intermittent gears are designed to allow the driven gear or follower one or 
more periods of rest during each revolution of the driver. This may be ac- 
complished in a rough manner by cutting out a number of teeth in the follower 
as illustrated in Fig. 192, but the cut and try method must be employed to 
obtain a definite ratio. This type of intermittent gear is seldom used, there 
being nothing but the spring b to keep the follower from moving during a 
period of rest, and the first tooth of the driver enters contact in a very un- 
certain manner, it sometimes being necessary to shorten the first tooth in 
the driver to prevent it from striking the top of the first tooth in the follower. 

The proper design of intermittent gears is not as difficult as it first appears. 
The pitch and outside diameters are found as for an ordinary spur gear, the 
pitch desired must correspond to an even number of teeth. The blank space 
on the driving gear is milled to the pitch line, and the stops in the follower 
are cut by a cutter of a diameter corresponding to the pitch diameter of the 
driver. If no such cutter is at hand, use the nearest to that size to rough 
out the stops and finish them with a fly cutter which can be set to any desired 
radius. 

It is well not to have the gears too near the same size ; the driver should be 
the smallest in order to secure all the contact possible in the stops. 

The simplest form of intermittent gear is shown by Fig. 193, the follower 
being moved but a short distance for each revolution of the driver. It will 
be noticed that a small amount of fitting will always be required at the point 
a to allow the point of the stop to clear. 

A more complicated drive is shown in Fig. 194, the follower being moved 
one-sixth of a revolution for each revolution of the driver. Each of these 
gears is turned up as for a spur gear of 30 teeth 5 diametral pitch. The cut- 
ting operation would be as follows: Index for 30 teeth; cut the first three teeth, 
then index for two teeth without cutting, and so on around the blank. The 
six stops are then milled, with a cutter 6 inches in diameter, to a depth of 
0.2 inch, or the addendum of the gear, which completes the follower. Four 
teeth are then cut in the driver, and the remainder of the blank milled to the 
pitch line. A little filing and the gears are complete. 

254 



INTERMITTENT GEARS 



255 



A still simpler method of cutting these gears, and one that avoids the neces- 
sity of first laying them out, is as follows: Drop a cutter equaling the pitch 
diameter of the driver into the blank of the follower, to the depth of the 
addendum at the points stops are desired. Then cut the first tooth at a 




American Machinist 
PIG. 193. 
INTERMITTENT GEARS WITH TWELVE STOPS. 



American Slachinist 



FIG. 194. INTERMITTENT GEARS WITH 
SIX STOPS. 



point midway between two of the stops, and continue cutting toward one 
of the stops until the point of the stop touching the outside circumference 
of the blank is cut away, or, in other words, until there is no blank space on 
the gear between the last space cut and the point of the stop. The same 
number of teeth are then cut in the opposite direction until the same 



256 



AMERICAN MACHINIST GEAR BOOK 



condition is met. If the stops are evenly 
spaced the cutting of the remaining teeth 
is a simple matter. If the stops are not 
evenly spaced the first tooth for each group 
must be located between each stop. The 
same number of teeth are then cut in the 
driver as there are spaces in the follower 
for each group and the remainder of the 
blank milled to the pitch line. For a pair 
of gears such as shown in Figs. 193 and 194, 
the cutting of the teeth by this process 
will be a simple matter. 
The cutting of internal intermittent gears is a counterpart of the above. 

Bevel gears, while being more difficult to cut, are governed by the same 

rules (see Fig. 195). 




American Machinist 



FIG. I95. INTERMITTENT BEVEL GEARS. 



MODIFICATIONS OF THE GENEVA STOP 

The accompanying engravings illustrate three highly ingenious and ex- 
tremely interesting modifications of the device used in watches to prevent 
overwinding which have been applied by Mr. Hugo Bilgram to various 
automatic machines constructed at his works. The constructions have a 
family resemblance in principle, though they are entirely unlike from a struct- 
ural standpoint. 

Three main features characterize the constructions: first, the intermittent 
motion of the Geneva stop; second, the entire absence of shock at engagement 
or disengagement; and third, the positive character of the movement — the 
parts being locked in position both when they are in motion and when they 
are idle. 

Fig. 196 represents the smallest departure from the watch mechanism. 
The interrupted disk a is the driver and revolves continually. The driven piece 
is seen at b, and the requirements are that the driven piece shall remain at 
rest during three-fourths of a revolution of the driver, and shall then make one- 
quarter of a turn during the remaining quarter turn of the driver. The driver 
may revolve in either direction, but supposing it to turn in the direction of 
the arrow, a roller c attached to the driver is about to enter one of four radial 
slots in the face of the driven piece. During the succeeding quarter turn of 
the driver the parts will move together, the motion of the follower ceasing 
when groove d has reached the position occupied in the figure by groove e, 
and this movement of the follower will obviously occupy ninety degrees of 



INTERMITTENT GEARS 



257 



angle. It will be seen that the parts are so laid out that roller c enters and 
leaves the grooves tangentially, insuring absence of shock at both the com- 
mencement and the conclusion of the engagement. 

At f g h i on the follower is a series of rollers raised above the faces sur- 
rounding the grooves, and the circular part of the driving disk carries a circular 
groove / k I at such a radial distance as to engage these rollers in succession 




FIG. 196. GENEVA STOP — MODIFICATION NO. I. 



during the idle period of the follower and hence lock it in position. It is 
obvious that in the direction of motion supposed, this circular groove is just 
leaving roller i, and so disengaging it preparatory to movement by roller c. 
On the completion of the follower's movement, roller/ will occupy the position 
of roller g in the illustration, while the end / of groove j k I will have turned 
to a position ready to embrace it and so lock the follower in position. With 
motion in the opposite direction, groove j k I in the position shown would be 
in the act of engaging roller i on the completion of the movement. Rollers 
g and i being at the same distance from the center m of the driver, both are 



258 



AMERICAN MACHINIST GEAR BOOK 



engaged by the groove during a revolution, the locking taking place with one 

and the unlocking with the other, both rollers being in the groove during most 

of the time. 
A modification of this gear, which it is unnecessary to show, has five grooves 

in the driven wheel with corresponding modifications in the character of the 

movement. 

In the second construction the motion of the follower is intermittent like 

the last, but with different relations between the idle and acting periods. 

The driver runs continuously, the relationship being: 
During 5-6 turn of the driver the follower is at rest. 
During 1 1-6 turn of the driver the follower makes one complete turn. 
In other words the follower makes one turn to every two turns of the driver, 

but this revolution of the follower occupies little more than a turn of the 

driver. 
Pinned to the face of the driven gear is the plate a, Fig. 197, the arm b of 

which is fitted to embrace a hub / on the driver shaft, whereby, until released, 

the follower is locked in its idle position. 
Revolving with this hub is an arm c car- 
rying a roller d, which is fitted to engage 
the slot g. As it does so, the notch e in 
hub / comes opposite finger h, thus dis- 
engaging the locking mechanism. Roller 
d enters tangentially without shock and 
accelerates the motion of the follower until 
the roller reaches the line of centers, when 
the gears engage and the motion goes on. 
The completion of the revolution of the 
driver finds the roller d again on the line of 
centers, but engaging slot i, and the con- 
tinuance of the motion brings the parts 
again to the positions shown. It should 
be noted that in this mechanism not only 
is the starting and stopping of the driver 
without shock, but at the instant the en- 
gagement of the gear teeth the roller d has 
brought the velocity of the follower up to 
that due to the gears, so that the transition 

of the motion from the pin to the gears and back again from the gears to the 

pin is also without shock. 

The most elaborate of these mechanisms is that shown in Figs. 198, 199, 




FIG. 197. 



GENEVA STOP — MODIFICA- 
TION NO. 2. 



INTERMITTENT GEARS 



259 



and 200. In this the driver — turning about a — is required to turn through 
about 73 per cent, of a revolution, while the follower stands still, the follower 
then making a complete revolution during the remaining 27 per cent, of the 
revolution of the driver. Figs. 198, 199, and 200 are side views intended to 
show the action in a succession of positions. 

The driver is an interrupted disk bed, Fig. 198, having cam-shaped 
edges e f at the mouth of the notch. Slightly in the rear of this disk is a 





FIG. 198. GENEVA STOP — MODIFICATION NO. 3. 



toothed sector g. The incomplete driven gear h meshes with g during the 
acting periods, and the purpose of the remainder of the mechanism is to start 
b in motion and throw the teeth in mesh as well as to lock the follower in posi- 
tion during the period that it stands still. Fig. 198 shows the parts in position 
at the beginning of the movement of the follower, which is still locked in the 
position which it occupies during its idle period. A bar i on the front face 
of gear h rides on, and up to this point has been locked in the idle position 



26o 



AMERICAN MACHINIST GEAR BOOK 



by the disk bed. Finger j — one of a pair j k — is attached to the rear 
of gear h, where it may turn freely between the sector g and the driving 
pulley /. This pulley I carries two rollers m n arranged to engage the 
ringers j k respectively. In the position of Fig. 198 the driving disk, moving 
in the direction of the arrow, has brought roller m into position, where it is 
about to engage ringer j, the direction of the acting side of j being tangential 
to the motion of m, so that the movement begins without shock. To permit 
j, b, and i to turn, the edge e of the disk is dressed off, but to such a degree 




FIG. 199. 



FIG. 200. 



GENEVA STOP — MODIFICATION NO. 3. 



that contact is maintained between the right-hand end of i and the edge 
of the disk as the follower turns, so that the motion is positive without slack. 
As the movement progresses the speed on the follower increases until the posi- 
tion of Fig. 199 is reached, when — the roller being on the pitch circle of sector 
g — the speed of the follower is the same as that due to the gears and the teeth 
drop into mesh without shock. From this on the gears drive, and the arms 
j k turn completely over, the position when the gears go out of mesh being 
shown in Fig. 200. From this on the action is the reverse of that shown by 
Figs. r98 and 200, the drhing piece being now the cam- shaped edge / of the 
disk, the ringer k preserving the positiveness of the motion and preventing 
the driven pieces overrunning by momentum as they are brought to rest, the 
final stopping being accomplished as the roller slips out of action in a tan- 
gential direction, and again without shock. From the position of Fig. 200 to 



INTERMITTENT GEARS 



261 



that of Fig. 198 the bar i simply rides on the edge of the driver disk and the 
follower remains at rest. 

" It should be remembered that these mechanisms are not models designed 
to embody a pretty movement invented beforehand, but they are parts of 
machines, some of which are made in considerable numbers, and have been 
devised, as occasion arose, to accomplish certain required results. As such, 
they represent the art of invention carried to a high degree of perfection." 




FIG. 20I. 



FIG. 202. 
AN INTERMITTENT SPUR GEAR. 



FIG. 203. 



The line cuts (Figs. 201, 202, and 203) show a pair of intermittent spur 
gears in three positions. The peculiarity about this gear is that although a 
dwell occurs, the teeth of the gear and pinion are in mesh at all times. 

The mutilated gear A is the driver and is secured to its shaft. A portion 
of its rim — dependent upon the length of dwell — is cut away. A segment 
B is mounted on the same shaft. This segment is free to swing in the cut- 
away portion of A , and is held in place against the side of A by a collar on the 
shaft. The teeth of B match with the teeth of A in both of its extreme posi- 
tions. B is held against the face C by a spring X. This spring is elastic 
enough to allow B to move as far as D. 

As the gear A moves in the direction 
of the arrow, it turns the pinion E. 
When B engages with E — the resistance 
of E being greater than the resistance 
of the spring X — the segment B re- 
mains stationary, while A moves till D 
comes in contact with B. During the 
time that B is at rest the pinion E is 
of course also at rest. As soon as D comes in contact with B, both B and E 
begin again to move. When B reaches a position where its teeth are no 
longer in engagement with the teeth of E, the spring X returns it to the face 




FIG. 204. AN INTERMITTENT WORM. 



262 



AMERICAN MACHINIST GEAR BOOK 



C. These gears were used as a feed gear for paper, the paper being cut 
during the dwell. 

The half-tone, Fig. 204, shows three views of an intermittent worm used 
in a looping machine. The pitch is 1-6 inch. The dwell is two-thirds of a 
turn, and the advance the remaining third. It was cut on an ordinary 16- 
inch lathe, using a mutilated change-gear. 



AN INTERESTING PAIR OF SPIRAL INTERMITTENT GEARS 

Figs. 205 and 206 show a pair of intermittent gears having the peculiar 
characteristics that, if the large gear be rotated continuously in one direction, 
the pinion will rotate alternately three-quarters of a turn in one direction and 
one-quarter of a turn in the opposite direction, with a rest or dwell between each 




FIG. 205. THE RIGHT-HAND TEETH IN 
MESH WITH THE "PEG" SEGMENT. 



FIG. 206. 



THE LEFT-HAND TEETH 
IN MESH. 



movement. Similar gears have been made as part of a certain machine, the 
nature of which we are not at liberty to mention. The angle of spiral of 
both gears is 45 degrees, and under ordinary conditions either gear could 
be the driver. The large gear, however, has around portions of its periphery 



INTERMITTENT GEARS 263 

two tongues — which are practically a continuation of certain of the teeth. 
These tongues fit into grooves in the pinion, and during their passage 
through the grooves lock the pinion at rest. Owing to this feature, the large 
gear must in this case be the driver. 

The pinion has twelve teeth, divided into four groups of three teeth each, 
with one of the before-mentioned slots between each group. Two opposite 
groups of teeth are cut left hand, the two alternate groups are cut both left 
and right, leaving the teeth like a series of pegs. 

, Imagine the handle at a position opposite the pinion, then the left-hand teeth 
in the large gear will be at the left. There are nine teeth cut in this segment 
which, when the handle is turned — in the direction of the hands of a clock — 
engage first with the three teeth of a double-cut group on the pinion, then 
with the three left-hand full teeth of the next group, then with the three 
teeth of the other double-cut group. The pinion has then turned three- 
quarters of a revolution. The tongue then engages with the slot in the pinion 
and the rotation of the pinion is arrested. The large gear turns until the three 
right-hand teeth on its periphery come into mesh with the double-cut group 
first referred to, and reverse the direction of rotation of the pinion for a space 
of three teeth, or one-quarter turn, when the tongue again locks the pinion 
and the handle reaches the starting position. The large gear has thus made one 
turn and the pinion has advanced through three groups of three teeth each, 
equal to three-quarters of a turn, and has reversed through one group of 
three teeth, or one-quarter turn. Thus the total advance is but two groups 
of three teeth, or one-half turn, and to bring the gear and pinion into the 
same relative position as they were at the start the wheel must make another 
complete turn. 

The blank for the large gear was turned to the extreme diameter across the 
top of tongue, mounted in the milling machine, and the left-hand teeth, which 
extend clear across the face, were gashed slightly below the level of the top 
of the tongue. The blank was then indexed halfway around from the 
central left-hand space, the table swung for right hand, and a deep right-hand 
gash made for locating. Then the blank was indexed halfway around from 
the central right-hand space, the table swung back for left hand, and the 
left-hand teeth section milled down to the proper gear diameter. Then the 
full-length left-hand teeth were cut, and also a gash made outside of the 
end teeth which join the tongue so as to get the curve of the tooth, but not 
going far enough to touch the tongue. The blank was then indexed back 
to the right-hand locating space; the table swung, blank milled down to the 
proper gear diameter, and the right-hand teeth were cut the same as the left- 
hand. The table was then set square, and an end mill was put into the 



264 



AMERICAN MACHINIST GEAR BOOK 



spindle, and the stock milled away — leaving the tongue. It will be noticed 
that it was not necessary to remount the blank at all. The outsides of the 
teeth joining the tongue were chipped out as smooth and true to curve as 
possible. 

An important feature was to get the angle of the end of the tongue just 
right, and not file it too far back, which would have allowed the large gear 
to carry the small one somewhat beyond the point where the tongue should 
enter the slot, thus causing the end of the tongue to strike the side of the 
small gear and arrest the movement or perhaps cause breakage. The cutting. 



^E^ 





FIG. 207. LAYOUT OF A PAIR OF INTERMITTENT SPIRAL GEARS. 



of the small gear was simply a matter of setting the cutter directly over the 
center of the blank and swinging the table for both hands; the slots were 
cut at the last machine operation, and then the little sharp projections which 
were left at the ends of the crosscut sections and near the slots were chipped 
off, as they were surplus and would interfere with the movement of the gears. 

Fig. 207 shows layout of a set of the same style of gearing as Figs. 205 and 
206, but of a different ratio. 

The gears were cut by the Boston Gear Works, and were prepared pre- 
liminary to cutting into more expensive blanks. The work was accomplished 
by the usual methods, and the finished gearing accomplished the desired 
results and is now in successful use. 



SECTION XI 
Elliptical Gears 

Elliptic gears are in general use on shapers, planers, slotters and similar 
machine tools to transmit a quick-return motion to the ram. The cost of 
production is more than for gears having circular pitch lines, but they are 
undoubtedly the cheapest quick-return motion among known mechanical 
movements. The method to be outlined is not new, but is as accurate as any 
and the cost of the tools is not high. This method is in general use in many 
shops to the exclusion of other methods not considered as good. 

In order more clearly to describe the process it will be well to take an actual 
case and carry it through from start to finish. Assume that a pair of gears 
is ordered to transmit a 3 to i quick-return motion and the centers are 8 
inches apart. About 3 diametral pitch is specified. 

METHOD OF LAYING OUT* 

Fig. 208 represents diagrammatically the method of laying out the gears. 
Lines A A' and B B' are first drawn perpendicular to each other. The major 
axis of the pitch line of the gear is the 
same as the given center distance, 8 
inches, and is laid off as shown. The 
focus points X X' are drawn in so that 
A X is in the same ratio to A' X as the 
given quick-return ratio. The points 
B and B f are located by setting the 
dividers at one-half the major axis A 
A' and cutting the minor axis with 
arcs having centers at X and X '. Arcs 
having radii equal to B' and A' are 
drawn covering one quadrant, as shown. 
This quadrant is divided into six equal 
divisions, as shown by radii from the 

center O. These radii are marked a, a 1 , a 2 , etc. Next intersecting lines are 
drawn from the intersection points of a, a 1 , a 2 , etc., with the circles, as shown, 

* W. E. Thompson. 
? 265 




FIG. 208. 



LAYOUT FOR AN ELLIPTIC 
GEAR. 



266 



AMERICAN MACHINIST GEAR BOOK 



and the intersection points of these perpendiculars are points on a perfect 
ellipse. 

" With a center on A A 1 find an arc that will very nearly cut the points from 
a, a 3 , a 2 , and A 1 . Also find a center on B B 1 about which an arc can be 
drawn cutting points B 1 , a, a 1 , and a 2 . These centers are used when cutting 
the teeth and should be laid out as accurately as possible. After rinding two 
centers and measuring the respective distances from 0, the other two centers 



iS, 



.rfn. 



gfflfor 



rbor 



n 



~z_ 



Drive Pin 



o 



•Vernier i ^ 

Vernier 



o 



\ 



rffassm. bB 




-f- j -+— m\— \-w- 
-4,\ i iji i i if 



End Elevation 



Side Elevation 



W 



Plan 



FIG. 209. FIXTURE FOR CUTTING ELLIPTICAL GEARS. 



may be put in and an ellipse drawn, as shown. This line is then stepped off 
with dividers set at the corrected tooth thickness for the desired pitch and the 
number of divisions noted. 

" It is preferable to have an odd number of teeth for convenience in cutting, 
so if the pitch line does not divide into an even number of divisions, half of 
which is an odd number, when the dividers are set to the chordal-tooth thick- 
ness of the desired pitch, the pitch line may be reduced or enlarged, as is 
found necessary. When the pitch is given, as it was in this case, the pitch 
line may be divided and the divider division measured. The corresponding 
pitch is used in selecting a cutter. In this case the line was divided into 30 
divisions which corresponded very close to 2M pitch, so these cutters were 
used. 

" Radial lines common to two centers of the elliptic arcs are drawn, as shown 
at b, b 1 , b 2 , and in the other quadrant not drawn. These lines are the dividing 
points between two different arcs and are drawn before laying out or cutting 
the teeth. Radial lines from the four centers are drawn, through the centers 
of the space divisions, as shown at c, c l , c 2 , etc. These lines are for the purpose 
of starting the first cut and checking the rest to prevent large errors. 

"The base circles of the two different curves are drawn, as shown, and a few 
teeth- laid out by an accurate odontograph on each curve, as shown, or cutters 
may be selected by measuring the pitch radii D A and E B and figuring the 
proper number of teeth to cut for. Cutters are selected by one or the other of 



ELLIPTICAL GEARS 



267 



these methods and the gear, previously having the shaft and driving pin holes 
X and X 1 bored and the addendum outline roughed, is ready for cutting. 

METHOD OF CUTTING 

" The fixture shown at Figs. 209 to 2 1 1 is bolted onto a circular milling attach- 
ment. The fixture consists of a base plate carrying two slides moving at right 
angles to each other. These slides are equipped with verniers registering zero 
when the center of the arbor is in line with the center of rotation of the milling 
attachment. The drive pin is riveted into a separate slide that is adjustable 
and is in line with the arbor and line of motion of the large slide. This fixture 
is placed in position on the machine and the rough-gear blank clamped in 
place. 

The point Z^Fig. 208, is then set over the center of rotation of the attachment 
by means of the verniers and the center of the cutter brought into the line A A 1 . 




FIG. 2IO. ELLIPTICAL GEARS WITH BORE IN CENTER, SHOWING SEPARATION OF TRUE 

ELLIPTICAL PITCH LINES. 



In this case the cutter is an end mill used to finish the outside of the gear. By 
moving around each quadrant setting the corresponding centers over the center 
of rotation for each quadrant the outside is milled true with the pitch line. 
A cutter for the ends is then put in the machine and in line with A A 1 . The 
first space may then be cut to depth with two or more cuts, depending on the 
tests for alignment. A piece of glass having a center line and a number of 
short lines scratched equal distances apart and from the center line is used 



268 



AMERICAN MACHINIST GEAR BOOK 



for testing the cuts. The center line is placed over the line A A 1 on the first 
cut, and the short lines brought intersecting the pitch line. By using a glass 
a variation of o.ooi inch may be readily seen and remedied. After the first 




FIG. 211. CUTTING THE TEETH. 



space is cut the gear-tooth calipers may be set and used to check the cutting. 
By checking with both the radial center lines and the tooth calipers errors are 
reduced to a minimum. After the first gear is cut it may be used as a templet 
for all others, or the indexing may be noted and repeated." 

Several years ago the author had an experience cutting elliptical gear that will 
be of interest. The driven gear was required to have two variations of speed 
per revolution; therefore the bore was put in the center of the gears instead 
of at the foci, which is usual. The gears cut are shown in Fig. 214. When 



ELLIPTICAL GEARS 



269 



the ellipse is very flat the four-arc method, described above, cannot be employed, 
as the pitch line cannot be described even approximately correct by four arcs. 
Therefore the gears were cut as illustrated by Fig. 211, the blank being set 
for each tooth cut. The teeth were first located on a templet which was 
secured to the face of one of the gears; they were both cut at one time. The 
blank was manipulated until the center line of each tooth space was brought 
in line with the center line of the cutter. 

This is accomplished with a surface gauge placed against the front ways of 
the milling machine as illustrated in Fig. 211. The depth of the teeth was 
obtained by first bringing the cutter to the outside of the templet and raising 





FIG. 212. THE GARDENERS' ELLIPSE. 



FIG. 213. THE METHOD EMPLOYED IN LAY- 
ING OUT GEARS SHOWN IN FIGS. 214 
AND 215. 



the table a distance equaling the depth of the tooth, or by locating the pitch 
line of the templet with another surface gauge from the cutter spindle. 

A cutter made to finish the outside diameter as the teeth are cut is a decided 
improvement over first milling or slotting the outer surface, although milling 
off the points of the teeth after they are cut is the next best plan. When the 
gears were mounted as shown in Fig. 210, it was found that the pitch lines 
separated at four points of the ellipse, as between b and c, therefore there was 
excessive backlash between the teeth at these points. 

The gears had been laid out with a gardeners' ellipse, using a piece of silk 
thread looped around pins set in the foci; the loop being adjusted until the 
describing point passed through the intersections of both the major and minor 
axes as shown in Fig. 212. Of course, we all thought the gardeners' ellipse was 
at fault, and it was decided to employ this process only on "tulip patches" 
in the future. 

A proper instrument was secured for the next attempt and everything done 
according to Hoyle. This time, instead of first cutting the gears and invest- 



270 



AMERICAN MACHINIST GEAR BOOK 



igating afterwards, two templets were laid out to represent the pitch lines 
of the gears. These templets developed the same error found in the first 
gears, so the pitch lines were corrected as per dotted lines in Fig. 210 and the 
gears w T ere satisfactory. 

Later on several gears of the same size were required and a search was made 
for a simpler method of laying them out. Noticing the attachment described 
by Mr. George B. Grant in his "Treatise on Gears," section 150, an attempt 




FIG. 214. ELLIPTICAL GEARS LAID OUT AS SHOWN IN FIG. 213. 



was made to apply this principle. A circle whose radius equalled the radii of 
both the major and minor axes was first drawn and divided into the number 
of teeth to be cut in the gear. The ellipse was then described from the same 
center. The center lines of the teeth were then projected from the points 
located on the outer circle to the ellipse, the edge of the blade being on a fine 
with the points e and/ as shown in Fig. 213. The center fine of the tooth 
thus projected did not always cross the pitch fine at right angles, except at 
the major and minor axes, and doubts were expressed as to the success of 
gears cut on these fines, but it was thought worth a trial at least. The pitch 
lines were corrected in the same manner as before and the gears were cut. 

They not only ran better than the first gears, but it was found necessary to 
remove part of the correction between the points b and c. This is accounted 
for by the fact that the circular thickness of the teeth increased between these 
points owing to the obliquity of the teeth; the thickness of the tooth being 
measured on the normal section (see Fig. 214). 



ELLIPTICAL GEARS 



271 



Gears with the bore located at the foci will not operate when cut in this man- 
ner ; the center lines of the teeth must be at right angles to the pitch line at all 
points. The interference of the teeth when thus mounted is shown in Fig. 215. 

When the bore is in the center of an elliptical gear it is better to have an 




FIG. 215. INTERFERENCE OF GEARS SHOWN IN FIG. 214, WHEN BORE IS AT FOCI. 



even number of teeth, otherwise the gears must be cut separately. For an odd 
number of teeth the tooth centered on the major axis must engage a space 
located at the minor axis of the engaging gear. It is better to use an even 
number and place the edge of the teeth on the major and minor axes, for gears 
of this type. 

For a complete treatment of the subject of elliptical gears Grant's " Treatise 
on Gear Wheels " is to be recommended. 



SECTION XII 

Epicyclic Gear Trains 

CALCULATIONS RESPECTING EPICYCLIC GEAR TRAINS 

DERIVATION OF FORMULAS FOR SEVERAL USUAL TYPES, AND EXTENSION OF THE 
METHOD OF ANALYSIS TO A SOMEWHAT COMPLEX EPICYCLIC TRAIN* 

This form of gearing, which is really that in which one gear revolves around 
the center of the one with which it is in contact, has received considerable at- 
tention, and one notices its use in several directions. We will therefore look 
into some of the calculations respecting it, leading from the simpler to the 
more complex. 

SIMPLE PALR OF GEARS IN FIXED BEARINGS 

Example I. 

If in Fig. 216 R and N are two gears in mesh, r and n being their respective 
numbers of teeth, their bearings being fixed, then: 

Velocity of drive N gear N r 
Velocity of drive R gear R n 9 

r 

or, N's velocity = R's velocity X — . 

n 

If, however, R revolves in a positive direction, n must revolve in the opposite, 
that is, in a negative direction. 

. ' . N's velocity = R's velocity X — . (1) 

n 

In all these calculations it is essential that great care be taken in order to 

obtain the correct sign of the resulting velocity. 

GEARS IN FLXED BEARINGS, WITH AN IDLER 

Example II. 

An intermediate gear I is placed in contact with both N and R, Fig. 217. 
The effect will be that of giving N motion in the same direction as R. 

y 

. * . N's velocity = R's velocity X — . (2) 

n 

* Francis J. Bostock. 

272 



EPICYCLIC GEAR TRAINS 



273 



^~^\ ^^f 





Fig. 216 Simple Pair of Gears 

in Fixed Bearings. 
Eq.l.N's V.=E's V. x-i 



R - N 

Fig.217 Gears in Fixed Bearings. 

with an Idler 
Eq.2.N's V. = R'b V. x-£- 




F 'R 

Fig. 21 8' .Simple Epicyclic Train 
Eq.3.N's V. =B's V..x U+-5-) 





'Fig. 219 Illustrating Rotation 

of N when it is Revolved 
about the Center of F. 



Fig.220 Second Stage in Deriving 
Equation 3:. Arm assumed to 
be fixed, F turned backward 



Fig. 221 Epicyclic Train 

with an Idler 
Eq.4JS's Y.=R's V. x (1—1) 




Fig. 222 Simple Epicyclic 

Train with Tnternal Gear 
Eq.S.N's Y.= R's V. x 




Fig.223 Internal Gear Train with inter- 
mediate Gear: the Arm Driving 
Eq.6 4 N*s V.= R's V. x (I4.-L) 



Fig. 224 Same Train as Fig.223 but 
with the Internal Gear Driving 





Eq.7.N's V. 



:R'sV. x (^£-p) 



Fig. 225 Compounded Gears in 

Fixed Bearings 
Eq.8.N's V. =R's V. x— ~ 



Fig. 226 Compounded Gears in 
Fixed Bearings 
See Equation 8 ; Fig. 225 




Fig. 227 Conipound Epicyclic Train 
Eq.9.N's V. =R's V. x (1— lEL) 







Fig. 228 Second Stage in Deriving Equation 9 
Arm assumed to be fixed, F turned 
backward 




Fig.229 Compound Epicyclic Train "with 
One Internal Gear 
.N'sV.= R*8 V. xd-r-I™-) 



Ea.10. 



FIGS. 2l6 TO 229. 
EPICYCLIC GEAR TRAINS WITH CORRESPONDING VELOCITY RATIO FORMULAS. 



274 



AMERICAN MACHINIST GEAR BOOK 



^% 




FIG.230. Compound Epicyclic Train -with 
Two Internal Gears 
See Eq.9,same as Fig. 227 



NOTATION 
K = Denotes Driving Gear, or, in some 

cases, Arm. 
r = Number of Teeth in Driving Gear. 
N= Denotes Driven Gear or Arm. 
n = Number of Teeth in Driven Gear. 
T, S and M denote Intermediate Gears. 
F = Denotes Fixed Gear. 
f = Number of teeth in it. 
V = Angular Velocity. 
//////// Denotes part which is fixed.. 




FIG.231 An Epicylic Train Consisting of Two 
Central Gears, One Arm carrying Two 
Planetary Gears, and Two Internal 
Gears, One of which is Fixed. 



FJG.232 Diagram of the Train of Fig 231 

Eq.ll.N's V.=E'sVx ( r'n-hrf ) 
Vn(r-i-iy 



FIGS. 230 TO 232. 
EPICYCLIC GEAR TEAINS WITH CORRESPONDING VELOCITY RATIO FORMULAS. 



SIMPLE EPICYCLIC TRAIN 

Example III. 

Two gears, F and N, are in mesh, the centers of which are on the arm R, 
which is capable of revolving around the center of F. It is required to find 
the velocity ratio between R and N when R revolves around the fixed gear F; 
Fig. 218 shows the arrangement. The gear N is subject to two motions due 
to the following two conditions: 

a. The fact of its being fixed to the arm R. 

b. The fact that it is in contact with the gear F. 

We will therefore in the first place suppose that they are not in gear, and 
that N cannot rotate on the arm R. Then if R makes one revolution around 
F it is obvious that N must also make one revolution around F, as in Fig. 219. 

: N's velocity, due to condition a, = R's velocity, 

the direction being the same as R's. 



EPICYCLIC GEAR TRAINS 275 

Secondly, if instead of R making one revolution around F in a + direction, 

we cause F to make one in the opposite, that is, negative direction, we shall 

have exactly the same effect. Therefore place F and N in mesh, and fix the 

arm R, as in Fig. 220. 

f 
Then if F makes — 1 revolution, N will make + — revolutions. (Accord- 

n 

ing to Equation 1.) 

But - 1 of F = + 1 of R. 

f 
. * \ 1 revolution of R = — revolutions of N, 

n 

f 
or, N's velocity due to conditions b = — R's velocity.' 

By addition we obtain the total impulses given in N, that is: 

f 
N's velocity = R's velocity + — R's velocity 

n 



= R's velocity ( 1 + J-Y (3) 



EPICYCLIC TRAIN WITH AN IDLER 

Example IV. 

If an intermediate gear / be inserted between F and N, as in Fig. 221, we 
have a similar case to the above; but the intermediate gear has the effect of 
changing the direction of revolution of N (Equation 2), due to its contact with 
F through /. 

. ' . N's velocity = R's velocity X ( 1 — — I (4) 

It will be seen that if f=n, N will not have any motion of rotation at all; and 
it will have a positive one if / < n and negative if / > n. Thus by the ad- 
justment of / and n one can obtain great reduction in speed by means of few 
moving parts. 

SIMPLE EPICYCLIC TRAIN WITH INTERNAL GEARS 

Example V. 

Instead of the driven gear N being external, it might have been internal, 
as shown in Fig. 222. The effect will be the same as inserting an intermediate 
gear in Example III, giving the same result as Case IV, namely: 

N's velocity = R's velocity X ( 1 - — V (s) 

In this case n > /. 

. * . The final direction is always + . 



276 AMERICAN MACHINIST GEAR BOOK 

INTERNAL GEAR EPICYCLIC WITH INTERMEDIATE GEAR 

Example VI. 

Fig. 223 shows a still further modification of this condition, / being an 
intermediate gear. The result is: 

N's velocity = R's velocity X ( 1 + — )• (6) 

THE SAME TRAIN WITH THE INTERNAL GEAR DRIVING 

Example VII. 

With the above type, one often arranges the outer internal gear to be the 
driver, imparting motion to the arm carrying the intermediate gear (see 
Fig. 224). 

We have seen by equation 6 that: 

N's velocity (driven) 1 

' ■ ' ■ T 

R's velocity (driver) 1 + 

n 



. ' . N's velocity = R's velocity -s- 1 1 — — J 



= R's velocity X (7377)- (7) 

The last two examples constitute what is known as the "Sun and Planet" 
gear, which is largely used in many mechanisms. All the above examples 
show "simple" gearing, but they can be compounded with great advantage. 

COMPOUNDED GEARS LN FLXED BEARINGS 

Example VIII. 

Gears compounded together are shown in Figs. 225 and 226, 226 being a 

diagram of 225. One repeats the well-known rule that: 

Velocity of driven gear Product of number of teeth of driver gears 

Velocity of driver gear Product of number of teeth of driven gears 

r ^^ m 
or, N's velocity = R's velocity X — — — . (8) 

s ?\ n 

The direction is the same as N's namely, + . 

COMPOUND EPICYCLIC TRAIN, WITHOUT INTERNAL GEAR 

Example IX. 

We will now arrange to fix one of the gears F, and by means of the arm R 
revolve the others around it, thereby causing N to revolve as shown in Figs. 
227 and 228. As before, we will assume the gears M and S to be out of mesh, 
so that when the arm R, carrying with it the gear X, makes one revolution 



EPICYCLIC GEAR TRAINS 277 

around F, N must also make one revolution relatively to F. Also when they 
are in mesh, the arm R being fixed and F makes one revolution in a negative 

direction (see Fig. 228), N will make — revolutions. (Equation 8.) 

o lb 

Now the total motion imparted to N must be the sum of these two, namely: 

f VYl 

i revolution of R = 1 — — ■ revolutions of N, 
J s n J 



or, 



(f X m\ 
1 —— — I. (9) 

S y\ n / 

COMPOUND EPICYCLIC TRAIN WITH ONE INTERNAL GEAR 

Example X. 

Fig. 229 shows a slight modification of the last case, N being an internal 
instead of an external gear. Obviously the only difference will be in the direc- 
tion of N's motion, that is: 

(f 7Yl\ 
I + J. (io) 

COMPOUND EPICYCLIC TRAIN WITH TWO INTERNAL GEARS 

Example XI. 

A further modification, however, is one in which both F and N are internal 
gears (Fig. 230), the effect of such being a change of sign in the equation. 

(f yyi \ 
I — 1. (9) 

The type shown in Figs. 227 and 230 is, perhaps, one of the best methods 
of obtaining a good reduction of speed in an easy and cheap manner. 

There are several combinations of the examples shown, but as they are all 
somewhat similar we will take another typical case as a guide for future calcu- 
lations. 

AN EPICYCLIC TRAIN CONSISTING OF TWO CENTRAL GEARS, ONE ARM 

CARRYING TWO PLANETARY GEARS, AND TWO INTERNAL GEARS, 

ONE OF WHICH IS FIXED 

Example XII. 

The writer has successfully used the arrangement shown in Figs. 231 and 
232, in which R and R' are two spur gears mounted on one shaft; / and V 
are two " planet" pinions, while F and N are two internal gears, the former 
being fixed. R and R r are made to revolve, which has the effect of giving N 
a very slow speed. 



278 AMERICAN MACHINIST GEAR BOOK 

A SCHEME FOR FINDING THE VELOCITY RATIO 

As this is somewhat complicated, we will work it out in stages: 

i. Obtain the revolutions of the arm A when R' makes one revolution, F, 
of course, being fixed. 

2. Obtain N's revolutions when the arm A is fixed and R makes one revo- 
lution. 

3. Assume R fixed, and that the arm makes one revolution; obtain, then, 
N's revolutions. 

4. Then if A T makes so many revolutions to one of the arm, as given by 
stage 3, we can by proportion obtain how many will be caused by the amount 
given by Stage 1 . 

5. Add the results of 2 and 4 together, and obtain the motion given to N 
by one revolution of R, which is the desired result. 

THE SCHEME WORKED OUT 

Working the above out we obtain: 

1. When F is fixed and R' makes one revolution, the arm A must make + 

(According to Equation 7.) 



r'+f 

T 

2. R makes one revolution, arm A being fixed; then A" must make 

n 

revolutions. (According to Equation 2. Negative sign used because of the 
internal gear.) 

3. When R is fixed and arm A makes one revolution, N will make + 1 1 -\ J 

revolutions. (According to Equation 6.) 

y 

4. With one revolution of arm, N makes 1 + — revolutions, from Stage 3; 

n 

r' 

. ' . with -, 7 revolutions of the arm, as derived in Stage 1, N will make 



hiH^il- 



/) 

5. The aggregate is the sum of the effects derived in Stages 4 and 2, namely, 
to one of R, N makes: 

{n + r) r' r 



(* + ~)x{?Tf) + (-i) = 



n (r f (Ef) n 
rr' + r' n — rr' — r f _ r' n — r f 
~ n ir' + /) ~ n (/ + f) ' 

The final direction of revolution of N will depend upon the relation which 



EPICYCLIC GEAR TRAINS 279 

/ n bears to r f; if the former be greater, then the direction will be positive 
(+), and vice versa. The formula for this combination is then: 

(Y "VI — Y f \ 
— i ' a- A )• ( i:e ) 

SOME NUMERICAL EXAMPLES IN EPICYCLIC GEARING 

In order to illustrate the above examples we will take one or two cases. 

If in Example and Fig. 218, / = 30, n = 25, then to one revolution of R, 

N will make H — — I = i + |f= 2\ revolutions. 

It will be obvious that with/ = n, N would revolve at twice the speed of R. 
In the type shown in Fig. 7,/ = 60, n = 65; 
then 

Velocity ofN = 1 -. -f __ 1 - |f _ _, j_ 
Velocity of R " 1 1 " 6T " 1S " 

The arrangement of Fig. 227 is much used. Let n = 60, / = 61, s = 40, 
m = 41. 

Then the velocity ratio between TV and R is 1 — - — : 1 

s n 

61 X 41 2501 . 

= 1 — — = 1 — — - — = say 1 : 24, m a minus direction. 

40 X 60 2400 

Illustrating Example XII, Fig. 231, let r = 90, / = 91,/ = 120, n = 121. 

Velocity of N r' n — r f 91 X 121 — 90 X 120 
Velocity of R n (r f + /) 121 (91 + 120) 

11,011 — 10,800 211 1 



121 X 211 121 X 211 121 



DIRECTION OF ROTATION OF GEARS 



The following, in reference to epicyclic gear calculations, is by Oscar J. 
Beale, American Machinist, July 9, 1908: 

" A very valuable article relative to epicyclic gears is by Prof. A. T. Woods 
in the American Machinist for February 14, 1889. This article is well- 
nigh perfect. It is so clear and comprehensive that it was of great help to 
me. I have read a number of later articles, and I have always gone back to 
this in order to clear up the subject. I think that many of your readers 
would like to see it reprinted. 

"About the best way to determine the direction of rotation of epicyclic gears 
is by careful inspection of the position of the members; then, if you make a 



28o 



AMERICAN MACHINIST GEAR BOOK 



Fixed 



FIG. 233. 























F 


L 













Arm 



DIAGRAM OF AN EPICYCLIC 
TRAIN 



correct statement of the effect of each 
member, Professor Woods' methods will 
bring the answer right. One must be 
sure to give the result the proper sign; 
that is, one must be able to add and to 
subtract algebraically. 

"I have sometimes used a sort of 
mental key in cases like this sketch, 
Fig. 233. If the pitch circle of L is 
smaller than the pitch circle of F, the 
rotation of L will be opposite that of 
the arm. If L is greater than F, the 
rotation of L will be the same as that 
of the arm. 

"This 'mental key' may help some; 
but after all, it is usually better to 

reason mathematically as in Professor Woods' article." 

We reprint herewith Professor Woods' article to which Mr. Beale refers 

because of its value in determining the direction of rotation of epicyclic trains. 

EPICYCLIC TRAINS* 

" An epicyclic train consists of a number of gear wheels, or pulleys, and belts, 
some of which are carried upon a revolving arm. For example, in Fig. 236 
the wheel F is fastened to the shaft B, about which arm A turns. This arm 
carries the axes of C and L, C being an idle wheel gearing with F and L. The 
motion of the wheel L is thus composed of three motions: (1) that which it 
has by reason of its revolution about B as a center, (2) that due to the revo- 
lution of the arm A about F, and (3) that due to its connection with F by 
means of the wheel C. We will consider the effect of these motions sepa- 
rately, and will begin with the simplest possible arrangement. In Fig. 234 
let A be an arm which revolves about a center B, and carries a wheel L, which 
we will suppose to be fastened to it. If the arm be turned through one revo- 
lution, the wheel L will in effect revolve once about its own center. This will 
be clear by an examination of the successive positions shown in dotted lines, 
the revolution of the arm being in the direction of the arrow. For example, 
follow the motion of a point such as P; at 1 it is to the right of the center, at 2 
below it, at 3 to the left, and at 4 above the center, finally returning to its first 
position on the right. We thus see that L has practically made one revolution 
about its own center, just as it would have if it had been fixed at B concentric 

* Prof. A. T. Woods. 



EPICYCLIC GEAR TRAINS 



281 



with the arm. If L has not revolved by reason of the revolution of A, the 
point P would have remained horizontally to the right of the center during the 
revolution of A. This is, of course, the same motion as that of a crank-pin 
and crank, and will be still more clear if we remember that, if the pin did not 
in effect revolve about its own center, it would not turn in the brasses, and they 
could be dispensed with. 

Now, considering the second motion of L, that due to the revolution of the 
arm about F, refer to Fig. 235, and let the wheel F be fixed, or a dead wheel con- 





Fig. 236 



Fig. 234 





Fig. 237 



DIAGRAMS OF EPICYCLIC GEARS 



centric with the arm A. Let F and L have the same number of teeth. Then 
while the arm revolves once in the direction indicated, L will revolve in the same 
direction, the result being the same as if the arm had remained fixed, and F had 
been revolved once in the opposite direction. The final motion of L, while the 
arm revolves once, is therefore one revolution, as in Fig. 234, supposing F to be 
removed, and one revolution, as in Fig. 235, supposing F to be in place and 
fixed; or, while the arm revolves once, L revolves twice in the same direction. 
Now, instead of F being fixed, let it revolve once in the direction of the dot- 



282 AMERICAN MACHINIST GEAR BOOK 

ted arrow. The effect of this will be to give one additional revolution to L, re- 
sulting in three revolutions of L to one of the arm. In order, then, to get at the 
resultant motion of the last wheel in epicyclic trains, we must consider the 
three independent motions separately: first, suppose the first wheel F, which is 
concentric with the arm, to be removed; second, suppose the first wheel to be in 
place and fixed; and third, suppose the arm to be fixed and the first wheel to 
revolve as intended. The final motion of L is the sum of these three motions. 
For the sake of brevity we will designate revolution in the direction of the 
hands of a watch ahead, or + ; and that in the opposite direction backward, 
or — . Thus expressed, the revolutions of L in Fig. 235, as we have just dis- 
cussed it, will be: 

(a) + 1 due to the revolution of the arm, 
+ 1 due to the revolution about F" , 
+ 1 due to the revolution of F. 

+ 3 revolution of L to one of the arm. 

As a further illustration assume that in Fig. 235, F has 40 teeth and L 30, 
then if F is a fixed wheel, L will revolve: 

(b) + 1 due to the revolution of the arm, 
+ ff due to the revolution about F, 

o due to the revolution of F, 



+ i revolutions of L to one of the arm. 

If F makes one revolution backward while the arm makes one ahead, we will 
have for L: 

(c) + 1 due to the revolution of the arm, 
+ || due to the revolution about F f 
+ f £ due to the revolution of F, 



+ V- revolutions of L to one of the arm. 

If F makes one revolution ahead, or in the same direction as the arm, the 
result is to balance the effect of the revolution about F, and we have for L: 

(d) + 1 due to the revolution of the arm, 
+ f i due to the revolution about F, 
— £§■ due to the revolution of F, 



+ 1 revolution of L to one of the arm, or the same as in Fig. 234. 
Similarly, if we take the conditions the same as (c) and let F 
have 30 teeth and L 40, we will have + 2H revolutions of L to 
one of the arm. 



EPICYCLIC GEAR TRAINS 283 

We will now consider the effect of introducing an idle wheel, as shown in 
Fig. 236. In the first place, let L equal F, and let F be a fixed wheel. The 
revolutions of L will be: 

(e) + 1 due to the revolution of the arm, 
— 1 due to the revolution about F, 
o due to the revolution of F, 



o revolution of L, or, in other words: a point P, which is, say, ver- 
tically over the center of L, will remain so throughout the revo- 
lution of the arm. If we assume the same condition as (a), the 
resulting revolution of L will be: 

(f) + 1 due to the revolution of the arm, 

— 1 due to the revolution about F, 

— 1 due to the revolution of F, 



— 1 revolution of L to one of the arm. 

If we let F have 40 teeth and L 30, and let F be a dead wheel, we will have 
for L: 

(g) + 1 due to the revolution of the arm, 

— j-J due to the revolution about F, 

o due to the revolution of F, 



— J revolution of L to one of the arm. 

Or, reversing the position of the wheels, making F = 30 and L = 40, the 
revolutions of L will be: 

(h) + 1 due to the revolution of the arm, 

— f $ due to the revolution about F, 

o due to the revolution of F, 



+ 

The arrangement shown at (e) is used in one form of rope-making machinery. 
The "arm" A, Fig. 236, is then the revolving frame which carries the bobbins 
on which the strands or wire have been wound. B is the center of this frame, 
and on it the wheel F is fixed. A small yoke or frame, which carries a bobbin, is 
fixed on the axis of L, there being as many of these wheels and bobbins as there 
are to be strands in the rope. Then if F and L have the same number of teeth 
as at (e), the axes of the bobbins always point in one direction, and the rope 
is laid up without twisting the separate strands. If L has a few less teeth 
than F, the strands will be given a slight twist, making the rope harder. 



284 



AMERICAN MACHINIST GEAR BOOK 



FM. 



Arrangements such as Figs. 235 and 236 are applicable to boring bars having 
sliding head. In such cases B would be the dead center on which the bar turns 
and on which the wheel F is fastened, being, therefore, a dead wheel. The 
wheel L is fastened to the end of the feed-screw in the bar, as shown at S, in 
Fig. 237, which represents the end view of the bar. While the arrangement 
is an epicyclic train, such as we have discussed, the explanation of it is extremely 
simple, because the motion to be determined is that of the screw 6* with regard 
to the bar, not with regard to the lathe, or any stationary object. 

As F is fixed, the effect on L of one revolution of the bar is the same as if the 
bar remained stationary and F revolved once. Thus, if F has 20 teeth and L 
40, the screw 5 will make, in the bar, f-jj- = 3^ of a revolution, while the 
bar revolves once. And if the screw has four threads to the inch, the feed will 
be Y A X M = y% inch. The effect of the idle wheel (shown in Fig. 237) is 
simply to change the direction of the feed. 

Another form of epicyclic train is that shown in Fig. 238 in which the last 
wheel is concentric with the arm and first wheel. This does not change the 

resultant motion of L in any way, but only 
L |i|[|[[ 1 illinium i irmi D makes a more convenient form for trans- 
mitting motion from L to other parts of 
the machine. If F and L have 40 and 30 
teeth, while the wheels C and D are equal, 
we will have the same motion as at (g) and 
(h), supposing F to be a dead wheel. A 
recent and novel application of this form 
of train is to be found in the Waterbury 
watch, the principle of which is shown in 
Fig. 239. In this figure a b is the face of 
the watch and c d the frame which carries 
the principal train of wheels, that from C 
to the balance wheel g. This frame turns 
about the center shown below it and the 
bearing in the face. It is driven by the 
spring e and carries the minute hand M, and hence revolves once each hour. 
Between the frame and the face is a pinion C, having 8 teeth, which is con- 
nected to the balance wheel g by the train of wheels shown, and itself gears 
with two wheels F and L, having, respectively, 44 and 48 teeth. F is 
fastened to the face, and so is a dead wheel, and L is fastened to the hour 
hand H by the tube, as shown. It remains to show that the hour hand will 
revolve once in 12 hours, as required by means of this connection. We have 
here an epicyclic train F C L in. which the first wheel is fixed. The revolu- 



Fig. 238 



-H 



-M 



~n — 

Fig. 239 



EPICYCLIC GEAR TRAINS 285 

tions of L during one revolution of the arm, as we have called it, or the 
frame c d are, therefore: 

+ 1 due to the revolution of the arm, 
— f f due to the revolution about F, 
o due to the revolution of F 



+ j s = + xV revolution during one revolution of the arm or minute hand, 
which is, of course, as it should be. The remainder of the train of wheels in 
this watch do not differ in principle from that ordinarily employed, a pecu- 
liarity being, however, that the entire " works," held in the frame c d, revolve 
within the case every hour. 

Another peculiar adaptation of epicyclic trains is for the production of very 
slow velocities, using a small number of wheels. For example, in Fig 238, 
let the numbers of teeth on the several wheels be F = 19, C = 20, D = 21, 
and L = 20, and let F be a dead wheel. Working out this train as we have the 
others, it will be found that, while the arm A makes one revolution, the last 
wheel / will make but j^-q of a revolution. In the same way, if we take the 
number of teeth in order as above, as 27, 40, 37 and 25, the last wheel will make 
but one revolution, while the arms makes 1000. 

We have chosen examples in which the first wheel is the dead wheel, as these 
are the simplest and most common. By adjusting the speed of the first wheel, 
however, it becomes possible to transmit velocities by means of epicyclic 
trains, which would be practically impossible by ordinary means. As an illus- 
tration, suppose it is required to have one shaft make 641 revolutions to one of 
another. As 641 is a prime number, this ratio could not be transmitted ex- 
actly by ordinary gearing on account of the large number of teeth required 
for a single wheel ; but by means of an epicyclic train it can be readily accom- 
plished. Of course, the necessity for such ratios as this rarely occurs in ma- 
chinery. 

A method of solving problems involving epicyclic trains, which will be more 
convenient for many than that which we have followed, is by means of a gen- 
eral formula. Let v = the value of the train of gears, or the product of the 
number of teeth on the drivers divided by the products of the numbers of teeth 
on the followers, which would be, in Fig. 238, 

FXD 
CXL. 

In case of pulleys, v = the product of the diameters of the drivers divided by 
the product of the diameters of the followers. Let /, / and a represent the 
number of revolutions of the first wheel, last wheel, and arm, respectively, in 
the same time. 



286 



AMERICAN MACHINIST GEAR BOOK 



Then, 



I — a 



If one direction is represented as + , the other will be represented as — . If 
the last wheel is to revolve in the same direction as the first, supposing the arm 
to be fixed, v is +, and if the opposite direction, it is — . For example, take 
the data as at (&); then, 

8 = 3~o = 1 whence / = -} a. 

o — a 

Again, let it be required that L shall make one revolution to iooo of A 

(Fig. 238), 

1 — 1000 . 999 27 X 37 F X D 



v = 



o — IOOO 



= + 



1000 



40 X 25 C X L ' 



TABLE OF PROPORTIONS OF DIFFERENTIAL BACK-GEARS 

The following table, originally published in American Machinist by Ernest 
J. Lees, gives data for ready reference. For a back-gear on drill presses and 



SIZE NO. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


n 


12 


Diameter of Pulley 


8 


8 


10 


10 


12 


12 


15 


15 


« 


15 


* 


18 


Face of Pulley 


3 


3 


3V2 


3V2 


4>S 


4Y2 


S« 


sH 


6M 


6M 


7 X A 


7^ 


Width of Belt 


2Y2 


2V 2 


3 


3 


4 


4 


5 


5 


6 


6 


7 


7 


Approximate H. P. at 
300 r. p. m. 


*y* 


2V 2 


4'4 


aK 


7H 


7H 


11 


11 


13 


13 


18 


18 


Shaft Diameter D 


T.V2 


iM 


i l A 


1Y2 


1% 


*y& 


iM 


1% 


2 


2 


3 


3 


Pitch Diameter pinion 
A 


2M 


AV2 


2) 


5r 


2* 


5} 


2* 


6 


3 


9 


5 


13 


Number of teeth in A 


18 


36 


18 


36 


18 


36 


18 


42 


18 


54 


15 


39 


Pitch Diameter Inter- 
nal Gear B 


9 


9 


io5 


IOf 


10? 


107 


12? 


12S 


18 


18 


25 


25 


Xumber of teeth in B 


72 


72 


72 


72 


72 


72 


OO 


00 


108 


108 


75 


75 


Pitch Diameter Idler C 


3% 


2\i 


,6 

3f 


2* 


37 


2* 


5? 


67 


7« 


AV2 


10 


6 


Xumber of teeth in C 


27 


18 


27 


18 


27 


18 


36 


24 


45 


27 


30 


18 


Xumber of Idlers 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


Diameter Pitch of 
Gears 


8 


8 


7 


7 


7 


7 


7 


7 


6 


6 


3 


3 


Face of Gears 


iM 


xV± 


iM 


i^ 


1^8 


T-Yi, 


iM 


iM 


2 


2 


3 


3 


Ratio 


5 
to 


3 
to 


5 
to 


3 
to 


5 
to 


3 
to 


6 
to 


3-142 

to 


7 
to 


3 
to 


6 
to 


2.923 
to 




1 


1 


1 


1 


1 


1 


1 


I 


1 


1 


1 


1 



TABLE 29. PROPORTIONS OF DIFFERENTIAL BACK-GEARS. 



EPICYCLIC GEAR TRAINS 



287 



other light machinery, the differential back-gear as originally designed. For a 
heavy drive and continuous service there is a better method of arrangement. 
This consists of using three idlers in place of one, these being equally spaced 
in order to retain the balance of the whole when locked up and driving direct. 
It will be readily seen that this arrangement calls for the following conditions 




FIG. 240. DIFFERENTIAL BACK-GEARING. 



Stationary 



in the gearing: That the number of teeth in pinion, idler, and internal gear 
must each be divisible by three, at the same time having correct diameters 
and pitch. 

A table is given herewith showing drives from 2}^ to 18 horse-power in 12 
sizes. This is arranged so that there are only 6 different diameters of internal 
gears, and by using different pinion and idlers one size can be used for two 
different ratios. 

Fig. 240 shows the principle on which these back-gears are operated. The 
locking device should not be copied, however, as it would be rather incon- 
venient to use, and as there are various methods of operating clutches the 
writer has not gone into detail on this point, but gives the general dimensions, 
leaving the rest to be worked out by the designer to suit the requirements and 
conditions of the machine on which it is to be used. 



SECTION XIII 

Friction Gears* 

A friction drive, as the term is here employed, consists of a fibrous or some- 
what yielding driving wheel working in rolling contact with a metallic driven 
wheel. Such a drive may consist of a pair of plain cylindered wheels mounted 
upon parallel shafts, or a pair of beveled wheels, or of any other arrangement 
which will serve in the transmission of motion by rolling contact. The use of 
such drives has steadily increased in recent years, with the result that the so- 
called paper wheels have been improved in quality, and a considerable number 
of new materials have been proposed for use in the construction of fibrous 
wheels. 

THE WHEELS TESTED 

Choosing materials which have been used for such purposes, driving wheels 
of each of the following materials have been tested: straw fiber, straw fiber 
with belt dressing, leather fiber, leather, leather faced iron, sulphite fiber, tarred 
fiber. 

The straw fiber wheels are worked out of the blocks which are built up usu- 
ally of square sheets of straw board laid one upon another with a suitable 
cementing material between them and compacted under heavy hydraulic 
pressure. In the finished wheel the sheets appear as disks, the edges of which 
form the face of the wheel. The material works well under a tool, but it is 
harder and heavier than most woods and takes a good superficial polish. The 
wheel tested was taken from stock. 

The wheel of straw fiber with belt dressing was similar to that of straw 
fiber, except that the individual sheets of straw board from which it was made 
had been treated, prior to their being converted into a block, with a "belt 
dressing" the composition of which is unknown to the writer. 

The leather fiber wheel was made up of cemented layers of board, as were 
those already described; but in this case the board, instead of being of straw 
fiber, was composed of ground sole leather cuttings, imported flax and a small 
percentage of wood pulp. The material is very dense and heavy. 

The leather wheel was composed of layers or disks of sole leather. 

* Abstract of paper presented to the American Society of Mechanical Engineers, December, 
1907, by W. M. Goss, Professor, University of Illinois. 



FRICTION GEARS 



289 



The leather faced iron wheel consisted of an iron wheel having a leather 
strip cemented to its face. After less than 300 revolutions the bond holding 
the leather face failed and the leather separated itself from the metal of the 
wheel. This wheel proved entirely incapable of transmitting power and no 
tests of it are recorded. 

The wheel of sulphite fiber was made up of sheets of board composed of 
wood pulp. The sulphite board is said to have been made on a steam-drying 
continuous process machine in the same way as is the straw board. 

The tarred fiber wheel was made up of board composed principally of tarred 
rope stock, imported French flax, and a small percentage of ground sole leather 
cuttings. 

• Each of the fibrous driving wheels was tested in combination with driven 
wheels of the following materials: iron, aluminum, type metal. All wheels 
tested, both driving and driven, were 16 
inches in diameter. The face of all driv- 
ing wheels was if inches while that of 
all driven wheels was J inch. 

The purpose of the experiments was to 
secure information which would permit 
rules to be formulated defining the power 
which may be transmitted by the various 
combinations of fibrous and metallic 
wheels already described. To accomplish 
this it was necessary to determine for each 
combination of driving and driven wheel 
the coefficient of friction under various 

conditions of operation; also the maximum pressures of contact which can be 
withstood by each of the fibrous wheels. 

The testing machine used is shown diagrammatically by Fig. 241. The 
principles involved will be made clear by assigning the functions of the actual 
machine to the several parts of this figure. The shaft A runs in fixed bearings 
and carries the fibrous friction wheel. This wheel is the driver. Its shaft A 
carries, besides the friction wheel, two belt pulleys, one on either side to which, 
from any convenient source of power, serve to give motion to the driver. The 
shaft B carries the driven wheel, which in every case was of metal. The bear- 
ings of this shaft are capable of receiving motion in a horizontal direction and 
by means of suitable mechanism connected therewith, the metal driven wheel 
may be made to press against the fibrous driver with any force desired. The 
pressure transmitted from B to A is hereinafter referred to as the "pressure 
of contact" and is frequently represented by the symbol P. The tangential 




///////>////////// 



FIG. 241. DIAGRAM OF TESTING MA- 
CHINE FOR FRICTION WHEELS. 



290 



AMERICAN MACHINIST GEAR BOOK 



forces which are transmitted from the driver to the driven wheel are received, 
absorbed and measured by a friction brake upon the shaft B. In action, there- 
fore, the driven wheel always works against a resistance, which resistance may 
be modified to any desired degree by varying the load upon the brake. The 
theory of the machine assumes that the energy absorbed by the brake equals 
that transmitted from the driver to the driven wheel at the contact point C. 




E 



FIG. 242. 



Partial Elevation 
THE TESTING iLACHTXE FOR FRICTION WHEELS. 



Accepting this assumption, the forces developed at the periphery of the brake 
wheel may readily be reduced to equivalent forces acting at the circumference 
of the driven wheel. The force, which is directly transmitted from the driver 
to the driven wheel, is hereinafter designated by the symbol F. It will be ap- 
parent from this description that the functions of the' apparatus employed are 
such as will permit a study of the relationship existing between the contact 
pressure P and the resulting transmitted force F, which relation is most con- 
veniently expressed as the coefficient of friction. It is, 



FRICTION GEARS 291 

It is obvious, in comparing the work of two friction wheels, that the one which 
develops the highest coefficient of friction, other things being equal, can be 
depended upon to transmit the greatest amount of power. 

The actual machine as used in the experiments is shown by Fig. 242. Its 
construction satisfies all conditions which have been defined except that shaft 
B, Fig. 241, does not run in bearings which are absolutely frictionless, as is 
required by a rigid adherence to the theoretical analysis already given. These 
bearings, however, are of the " standard roller bearing" type and of ample size, 
and it is believed that the friction actually developed by them is so small 
compared with the energy transmitted between the wheels that it may be 
neglected. 

The bearings of the fixed shaft A are secured to the frame of the machine. 
The bearings of the axle B are free to move horizontally in guides to which 
they are well fitted. Those bearings are connected by links to the short arm of 
a bell crank lever, the arm of which projects beyond the frame of the machine 
at the right-hand end and carries the scale pan and weights E. The effect of 
the weights is to bring the driven wheel in contact with the driver under a 
predetermined pressure, the proportions of the bell crank lever being such as 
to make this pressure in pounds equal, 

P = 10 W + 73, 

where W is the weight on the scale pan E. 

The fulcrum of the bell crank lever is supported by a block G which may be 
adjusted horizontally by the hand wheel H at the rear of the machine, so that 
whatever may be the diameter of the driven wheel, the long arm of the bell 
crank may be brought to a horizontal position. The constants employed in 
calculating the coefficient of friction from observed data are as follows: 

Diameter of friction wheels (inches) 16 

Effective diameter of brake (inches) 18.35 

Ratio of diameter of friction wheel to that of brake wheel. 1.145 
Effective load on brake F' 

F' 
Coefficient of friction i-i45 — 5- 

The slippage between the friction wheels was determined from the readings 
taken from the counters connected to each one of the shafts. 

THE TESTS 

In proceeding with a test, load was applied to the scale pan E, Fig. 242, to 
give the desired pressure of contact, after which the hand wheel H at the back 
of the machine was employed to bring the bell crank to its normal position. 



292 



AMERICAN MACHINIST GEAR BOOK 



This accomplished, with the driving wheel in motion, the driven wheel would 
roll with it under the desired pressure of contact. A light load was next placed 
upon the brake to introduce some resistance to the motion of the driven shaft, 
and conditions thus obtained were continued constant for a considerable period. 
Readings were taken simultaneously from the counters and time noted. After 
a considerable interval the counters were again read, time again noted, and the 
test assumed to have ended. From the readings of the counters and from the 
known diameters of the wheels in contact, the percentage of slip attending the 
action of the friction wheels was calculated. Three facts were thus made of 



































R 




















































,0 













0.4 
















■ 3 
























Pr^ 


^6 




















0.3 








/ 






























If 
























0.2 






/ 






























/ 


























0.1 












St 1 


•a'tt 


Fi 


ber 


Ir< 


»n 


















l 


50 Pou 


idsj Pr 


ess 


ire 







































































0.5 






























• 
































i 

o 0.4 

•a 
fa 

o 0.3 

S3 






































































/ 




















































1 0.2 






























































0.1 












Stijaw 


Fi^er 


Iron 


















4 


00 1 


^ou 


adsj Pi 


essure 


























| 















Slip, Percent 
FIG. 243 



Slip, Percent 
FIG. 244 



CURVES FOR STRAW FIBER AND IRON, TYPICAL FOR ALL CURVES PLOTTED FROM THE 

FRICTION TESTS 



record, namely: (a) The pressure of contact, (b) the coefficient of friction de- 
veloped, and (c) the percentage of slip resulting from the development of said 
coefficient of friction. 

This record having been completed, the load upon the brake was increased 
and observations repeated, giving for the same pressure of contact a new co- 
efficient of friction and a higher percentage of slip. This process was con- 
tinued until the slippage became excessive and in consequence thereof the 
rotation of the driver ceased. By this process a series of tests was developed 
disclosing the relation between slip and coefficient of friction for the pressure 
in question. Such a series having been completed, the load upon the weight 
holder E was changed, giving a new pressure of contact, and the whole process 
repeated. As the work proceeded, curves showing the relation of coefficient 
of friction and slip for pressures per inch width of face in contact of 1 50 pounds 
and 400 pounds, respectively, were secured. The curves shown by Figs. 243 
and 244 for the straw fiber driving wheel in contact with the iron driven wheel 
are typical in their general form of those obtained from all combinations of 



FRICTION GEARS 



293 



wheels, but the curves of no two combinations were alike in their numerical 
values. 

Having completed this series of tests at constant pressure, a series was next 
run for which the coefficient of slip was maintained constant at 2 per cent., and 
the pressure of contact varied from values which were low to those which are 
judged to be near the maximum for serv- 
ice conditions, with the result, which in 
all cases were similar in character with 
those given for the straw fiber and iron 
wheels, as set forth by Fig. 245. The 
numerical values of the points for other 
combinations were not the same as those 
shown by Fig. 245, but in the case of 
most of the combinations the coefficient 
of friction at constant slip gradually 
diminishes as the pressure of contact is 
increased. 

As the series of tests involving each 
combination of wheels proceeded, the increase in pressure of contact was dis- 
continued when the markings made upon the driving wheel by the metallic 
follower became so distinct as to suggest that a safe limit had been reached; 
but when all other data had been secured, tests were run for the purpose of 
determining the ultimate resistance of the fibrous wheel to crushing. The 
details of these will be described later. 



































0.5 

d 






























































































£0.4 






















— = ~Y — 








«M 0.3 
































O 
































a 

•3 0.2 

ft 






























































O0.1 










Straw-Fiber Iron 


















2 Percent; Slip 













































150 



200 250 300 350 400 
Contact Pressure 



PIG. 245. CURVE FOR STRAW FIBER AND 
IRON WITH CONSTANT SLIP 



COEFFICIENT OF FRICTION DEVELOPED BY THE SEVERAL COMBINATIONS OF 

WHEELS — STRAW FIBER AND IRON 

The results of experiments involving a straw fiber driver and an iron driven 
wheel are shown graphically in Figs. 243, 244, and 245. Figs. 243 and 244 il- 
lustrate the relation between slip and coefficient of friction when the two wheels 
are working together under pressures per inch width of 150 and 400 pounds, 
respectively. 

The figures show that although the values of the coefficient of friction are 
slightly lower than corresponding ones for 150 pounds pressure, the curves 
are sufficiently similar to establish the fact that the law governing change in 
coefficient friction with slip is independent of the pressure of contact. When the 
slippage is 2 per cent, the coefficient of friction is 0.425 for a contact pressure of 
400 pounds. That the coefficients of friction for all pressures between the limits 
of 150 pounds and 400 pounds are practically constant is well shown by the 
diagram Fig. 245. The pressure of 400 pounds is the maximum at which tests 



294 AMERICAN MACHINIST GEAR BOOK 

of this combination of wheels were run, though straw fiber was successfully 
worked up to a pressure of 750 pounds. 

STRAW FIBER AND ALUMINUM 

By curves plotted from values for a straw fiber driver and aluminum driven 
wheel, it can be shown that when the working pressure is 150 pounds per inch 
width and the slippage is 2 per cent, the coefficient of friction is 0.455; a l so > 
that for all pressures ranging from 100 to 400 pounds, the coefficient of 
friction is practically constant when the rate slip is constant. The maximum 
pressure at which tests involving this combination of wheels were run was 400 
pounds per inch width. 

STRAW FIBER AND TYPE METAL 

By curves plotted from values for a straw fiber driver and a type metal 
driven wheel it can be shown that when the two wheels are operated under a 
pressure of contact of 150 pounds per inch width and when the slip is 2 per cent. 
the coefficient of friction is 0.310; also, that for all pressures of contact ranging 
from 100 to 400 pounds, the coefficient of friction is practically constant when 
the slip is constant. 

STRAW FIBER WITH BELT DRESSING AND IRON 

Curves plotted from values for a straw fiber driver treated with belt dressing, 
and an iron driven wheel show that when the two wheels are worked together 
under a pressure of 1 50 pounds per inch width and when the slip is 2 per cent. 
the coefficient of friction is 0.12 ; also, that for all pressures up to 400 pounds per 
inch width, the coefficient of friction remains constant. The greatest pressure 
at which tests of this combination of wheels were run was 500 pounds per inch 
width. 

LEATHER FIBER AND IRON 

Curves plotted from the results of tests involving a leather fiber driver and 
an iron driven wheel show that when the two wheels are worked together under 
pressure of 150 pounds per inch in width and when slip is 2 per cent, the co- 
efficient of friction is 0.515. When the contact pressure is 300 pounds per inch 
width, the coefficient of friction is 0.510. The greatest pressure at which tests 
of this combination of wheels were run was 350 pounds per inch width, al- 
though leather fiber was successfully worked up to a pressure of 1200 pounds 
per inch width. 

LEATHER FIBER AND ALUMINUM 

Curves plotted from the results of experiments involving a leather fiber 
driver and an aluminum driven wheel show that under a contact pressure of 150 
pounds per inch width and a slip of 2 per cent, the coefficient of friction is 0.495. 



FRICTION GEARS 295 

This value remains practically constant under all pressures. The maximum 
pressure used in tests of this combination of wheels was 400 pounds. 

LEATHER FIBER AND TYPE METAL 

Curves plotted from the results of experience involving a leather fiber driver 
and a type metal driven wheel show that when the wheels are operated under 
a contact pressure of 150 pounds per inch width and when the slip is 2 per cent, 
the coefficient of friction remains constant for all pressures up to 400 pounds 
per inch width. 

TARRED FIBER AND IRON 

Curves plotted from the results of the experiments involving a tarred fiber 
driver and an iron driven wheel show that the change in the value of the co- 
efficient of friction with change of slip is practically independent of the pressure 
of contact. When the slip is 2 per cent., the coefficient of friction is 0.2 20 for a 
pressure of contact of 150 pounds and 0.250 for a pressure of contact of 400 
pounds per inch width. 

Tests of this combination were made also under different speeds when the 
wheels were working together under a pressure of contact of 250 pounds per 
inch width and when the slip was 2 per cent., with the result that the coefficient 
of friction was found to remain nearly constant for speeds of 450 and 3350 feet 
per minute, respectively. The greatest pressure at which tests of this combi- 
nation of wheels were run was 400 pounds per inch width, although tarred 
fiber was successfully worked up to a pressure of 1 200 pounds per inch width. 

TARRED FIBER AND ALUMINUM 

Curves plotted from the results of experiments involving a tarred fiber 
driver and an aluminum driven wheel show that when the slip is 2 per cent, 
and the pressure of contact 150 pounds per inch width, the coefficient of friction 
is 0.305 ; also, that for a pressure of 400 pounds per inch width, the coefficient 
of friction is 0.295. The greatest pressure at which tests of this combination 
were run was 400 pounds per inch width. 

TARRED FIBER AND TYPE METAL 

Curves plotted from the results of experiments involving a tarred fiber driver 
and a type metal driven wheel show that when the slip is 2 per cent, the co- 
efficient of friction developed under 150 pounds pressure per inch width is 
0.275; and under 400 pounds pressure per inch width, the coefficient of friction 
is 0.270. The maximum pressure at which tests of this combination of wheels 
were run was 400 pounds per inch width. 



296 AMERICAN MACHINIST GEAR BOOK 

LEATHER AND IRON 

Curves plotted from the results of experiments involving a leather driver 
and an iron driven wheel show that when the slip is 2 per cent, the coefficient 
of friction under a pressure of contact of 150 pounds per inch in width is 0.225 
and under a pressure of 400 pounds, 0.215. The maximum pressure at which 
tests of this combination of wheels were run was 400 pounds per inch width, 
although the leather driver was successfully operated up to a pressure of 750 
pounds per inch width. 

LEATHER AND ALUMINUM 

Curves plotted from the results of experiments involving a leather driver and 
an aluminum driven wheel show that when the pressure is 150 pounds per inch 
in width and the slip is 2 per cent, the coefficient of friction is 0.260; and when 
the pressure is 300 pounds per inch in width, the coefficient of friction is 0.295. 
The maximum pressure at which tests of this combination of wheels were made 
was 350 pounds per inch width. 

LEATHER AND TYPE METAL 

Curves plotted from the results of the experiments involving a leather driver 
and a type metal driven wheel show that when the slip is 2 per cent, and the 
contact pressure 150 pounds per inch width, the coefficient of friction devel- 
oped is 0.410. The greatest pressure at which tests of this combination of 
wheels were run was 350 pounds per inch width. 

SULPHITE FIBER AND IRON 

Curves plotted from the results of the experiments involving a sulphite fiber 
driver and an iron driven wheel show that when the slip is 2 per cent, and the 
pressure 150 pounds per inch width, the coefficient of friction is 0.550. The 
maximum pressure at which tests of this combination of wheels were run was 
350 pounds per inch width, although the sulphite fiber wheel was successfully 
operated up to a pressure of 700 pounds per inch width. 

SULPHITE FIBER AND ALUMINUM 

Curves plotted from the results of the experiments involving a sulphite fiber 
driver and an aluminum wheel show that when the slip is 2 per cent, and the 
pressure 150 pounds per inch width, the coefficient of friction developed is 0.410. 
The greatest pressure used in tests of this combination of wheels was 350 
pounds per inch width. 

SULPHITE FIBER AND TYPE METAL 

Curves plotted from the results of the experiments involving a sulphite fiber 
driver and a type metal driven wheel show that when the slip is 2 per cent, and 



FRICTION GEARS 297 

the contact pressure 150 pounds per inch width, the coefficient of friction is 
0.515. The maximum pressure used in tests of this combination of wheels was 
350 pounds per inch width. 

RESISTANCE TO CRUSHING 

Upon the completion of tests designed to disclose the frictional qualities of 
the several combinations, each fibrous wheel was subjected to test for the pur- 
pose of determining the maximum pressure per inch width of the face which 
could be sustained by it. This was accomplished by placing the wheel to be 
tested in the machine under a pressure of contact of 200 pounds per inch width. 
The load on the brake was then adjusted to give a 2 per cent, slip, and this 
brake load was maintained without change throughout the remainder of the 
tests. Thus adjusted, the machine was operated until the driver had completed 
15,000 revolutions. This accomplished, and for the purpose of determining 
the reduction, if any, in the diameter of the fibrous wheel, the brake load was 
removed and the operation of the machine continued without load for a period 
of 6000 revolutions, the readings of the counters being taken at the beginning 
and at the end of the period. Under conditions of no load, the actual slip was 
assumed to be zero and the apparent slip observed was used for determining 
the reduction in diameter of the fibrous wheel which had been brought about 
by the previous running under pressure. This accomplished, the pressure of 
contact was increased, usually by 100 pound increments, and the whole opera- 
tion repeated. This process was continued until failure of the fibrous wheel 
resulted. It will be seen that the ultimate resistance to crushing, as found by 
the process described, is that pressure which could not be endured during 15,000 
revolutions. 

A summary of results is as follows: 

A CONCLUSION AS TO METAL WHEELS 

An examination of Table 30, which presents a comparison of values represent- 
ing the coefficient of friction of the several combinations of wheels tested, re- 
veals the fact that the relative value of the metal driven wheels is not the same 
when operated in combination with different fibrous driving wheels. It appears 
that those driving wheels which are the more dense work more efficiently with 
the iron follower than with either the aluminum or type metal followers; but 
in the case of the softer and less dense driving wheels, and especially in the case 
of those in which an oily substance is incorporated, driven wheels of aluminum 
and type metal are superior to those of iron. Finely powdered metal which is 
given off from the surface of the softer metal wheels seems to account for this 
effect, and the character of the driving wheels is perhaps the only factor neces- 



298 AMERICAN MACHINIST GEAR BOOK 

sary to determine whether its presence will be beneficial or detrimental. 
Finally, with reference to the use of soft metal driven wheels, it should be 
noted that no combination of such wheels with a fibrous driver appears to have 
given high frictional results. Except when used under very light pressures, 
the wear of the type metal was too rapid* to make a wheel of its material serv- 
iceable in practice. 

CONCLUSIONS AS TO FIBROUS WHEELS 

The relative value of the different fibrous wheels when employed as drivers 
in a friction drive may be judged by comparing their frictional qualities as set 
forth in Table 30 and their strength as set forth in Table 31. The results show 
at once that the addition of belt dressing to the composition of a straw fiber 
wheel is fatal to its frictional qualities. The highest frictional qualities are 
possessed by the sulphite fiber wheel, which, on the other hand, is the weakest 
of all wheels tested. The leather fiber and tarred fiber are exceptionally strong; 
and the former possesses frictional qualities of a superior order. The plain 
straw fiber, which in a commercial sense is the most available of all materials 
dealt with, when worked upon an iron follower possesses frictional qualities 
which are far superior to leather, and strength which is second only to the 
leather fiber and the tarred fiber. 

THE POWER CAPACITY OF FRICTION GEARS 

A review of the data discloses the fact that several of the friction wheels 
tested developed a coefficient of friction which in some cases exceeded 0.5. 
That is, such wheels rolling in contact have transmitted from driver to driven 
wheels a tangential force equal to 50 per cent, of the force maintaining their 
contact. These wheels also were successfully worked under pressures of con- 
tact approaching 500 pounds per inch in width. Employing these facts as a 
basis from which to calculate power, it can readily be shown that a friction 
wheel a foot in diameter, if run at 1000 revolutions per minute, can be made to 
deliver in excess of 25 horse-power for each inch in width. It is certainly true 
that any of the wheels tested may be employed to transmit for a limited time 
an amount of power which, when gauged by ordinary measures, seems to be 
enormously high; but obviously, performance under limiting conditions 
should not be made the basis from which to determine the commercial capacity 
of such devices. In view of this fact, it is important that there be drawn from 
the data such general conclusions with reference to pressures of contact and 
frictional qualities as will constitute a safe guide to practice. 



FRICTION GEARS 



299 



Sulphite Fiber 

Leather Fiber 

Straw Fiber 

Tarred Fiber 

Leather 

Straw Fiber with belt dressing. 



COEFFICIENT OF FRICTION WHEN CONTACT 
PRESSURE IS 150 POUNDS PER INCH 



IRON 


ALUMINUM 


TYPE METAL 


0.550 


0.530 


0.5I5 


0.5I5 


0.495 


0.350 


O.425 


0.455 


O.31O 


O.250 


0.305 


O.275 


O.225 


O.360 


O.41O 


O.I20 







Table 30— Coefficient of Friction. 





load in 


DECREASE IN 






pounds 


DIAMETER 




Straw Fiber •< 


200 
650 

75o 


O.OOO 
O.053 
O.I25 


I Wheel failed before running 15,000 revolutions 
under 750 pounds pressure. 


r 


200 


O.OOO 


■> 




300 


O.OO5 






400 


O.OI3 






500 


0.02I 




Leather Fiber < 


600 
700 
800 


O.027 
O.04O 
O.051 


> Wheel failed before running 15,000 revolutions 
under 1200 pounds pressure. 




900 


O.068 






1000 


O.O99 






1 100 


O.I25 




L 


1200 


0.200 


> 


c 


200 


O.OOO 


> 




300 


O.O26 






400 


O.O38 






500 


O.052 




Tarred Fiber < 


600 
700 
800 


O.071 
O.O98 
O.138 


j, Wheel failed before running 15,000 revolutions 
under 1200 pounds pressure. 




900 


O.182 






1000 


O.250 






1 100 


O.295 




L 


1200 




J 


r 


35o 


O.047 


■> 


Leather 


45o 
55o 
650 


0.090 
O.OI5 
O.24O 


j.Wheel failed before running 15,000 revolutions 
under 750 pounds pressure. 


L 


75o 




J 




200 


O.OIO 


- 




300 


0.032 




Sulphite Fiber < 


400 
500 


0.056 

0.088 


> Wheel failed before running 15,000 revolutions 
under 700 pounds pressure. 




600 


0.146 






7°o 


0.258 


J 



Table 31— Strength of Various Fiber Wheels. 



300 AMERICAN MACHINIST GEAR BOOK 

WORKING PRESSURE OF CONTACT 

The results of these experiments do not furnish an absolute measure of the 
most satisfactory pressure of contact for service conditions. Other things being 
equal, the power transmitted will be proportional to this pressure, and hence it 
is desirable that the value be made as high as practicable. On the other hand, 
it has been noted as one of the observations of the test that as higher pressures 
are used, there appears to be a gradual yielding of the structure of the fibrous 
wheels; and it is reasonable to conclude that the life of a given wheel will in a 
large measure depend upon the pressure under which it is required to work. 
After a careful study of the facts involved, it has been determined to base an 
estimate of the power which may be transmitted upon a pressure of contact 
which is 20 per cent, of the ultimate resistance of the material as established 
by the crushing tests already described. This basis gives the following results: 

SAFE WORKING PRESSURES OP CONTACT 

PRESSURE 

Straw fiber 150 

Leather fiber . 240 

Tarred fiber 240 

Sulphite fiber 140 

Leather ( 150 

COEFFICIENT OF FRICTION 

The coefficient of friction for all wheels tested approaches its maximum value 
when the slip between driver and driven wheel amounts to 2 per cent, and, 
within narrow limits, its value is practically independent of the pressure of con- 
tact. A summary. of maximum results is shown by Table 30. In view of these 
facts, it is proposed to base a measure of the power which may be transmitted 
by such friction wheels as those tested upon the frictional qualities developed 
at a pressure of 150 pounds per inch of width, when operating under a load 
causing 2 per cent. slip. For safe operation, however, deductions must be 
made from the observed values. Thus, the results of the experiments disclose 
the power transmitted from wheel to wheel, while in the ordinary application 
of friction drives some power will be absorbed by the journals of the driven 
axle so that the amount of power which can be taken from the driven shaft will 
be somewhat less than that transmitted to the wheel on said shaft. Again, 
under the conditions of the laboratory, every precaution was taken to keep the 
surfaces in contact free of all foreign matter. It was, for example, observed 
that the accumulation of laboratory dust upon the surfaces of the wheels had 
a temporary effect upon the frictional qualities of the wheels, and friction wheels 



FRICTION GEARS 301 

in service are not likely to be as carefully protected as were those in the labora- 
tory. In view of these facts, it has been thought proper to use as the basis from 
which to determine the amount of power which may be transmitted by such 
wheels as those tested, a coefficient of friction which shall be 60 per cent, of 
that developed under the conditions of the laboratory. This basis gives the 
following results: 

COEFFICIENT OF FRICTION WORKING VALUES 

COEFFICIENT 
OF FRICTION 

Straw fiber and iron °- 2 55 

Straw fiber and aluminum 0-273 

Straw fiber and type metal 0.186 

Leather fiber and iron 0.309 

Leather fiber and aluminum 0.297 

Leather fiber and type metal 0.183 

Tarred fiber and iron 0.150 

Tarred fiber and aluminum 0.183 

Tarred fiber and type metal 0.165 

Sulphite fiber and iron 0.330 

Sulphite fiber and aluminum 0.318 

Sulphite fiber and type metal 0.309 

Leather and iron o- 3 ^ 

Leather and aluminum 0.216 

Leather and type metal 0.246 

HORSE-POWER 

Having now determined a safe working pressure of contact and a represent- 
ative value for the coefficeint of friction, it is possible to formulate equations 
expressing the horse-power which may be transmitted by each combination 
of wheels tested. Thus, calling d the diameter of the friction wheel in inches, 
W the width of its face in inches, and N the number of revolutions per minute, 
the equations become, for combinations of, 

HORSE-POWER 

Straw fiber and iron , . o. 00030 dWN 

Straw fiber and aluminum o . 00033 dWN 

Straw fiber and type metal. . '. o. 00022 dWN 

Leather fiber and iron o . 00059 dWN 

Leather fiber and aluminum o. 00057 dWN 

Leather fiber and type metal o . 00035 dWN 

Tarred fiber and iron o . 00029 dWN 



302 AMERICAN MACHINIST GEAR BOOK 

HORSE-POWER 

Tarred fiber and aluminum o. 00035 dWN 

Tarred fiber and type metal o. 00031 dWN 

Sulphite fiber and iron o. 00037 dWN 

Sulphite fiber and aluminum o . 00035 dWN 

Sulphite fiber and type metal o. 00034 dWN 

Leather and iron o . 00016 dWN 

Leather and aluminum o. 00026 dWN 

Leather and type metal o. 00029 dWX 

The accompanying chart gives a convenient means of determining the value 
of any one of the variable factors in the formula horse-power = 0.0003 dWN 
for the straw fiber friction wheel working in combination with an iron follower, 
the remaining factors being known or assumed. To transform values thus 
found to corresponding ones for the other possible combinations of wheels, 
it is necessary only to multiply by the proper factor chosen from the table of 
multipliers given with the chart. 

APPLICATION OF RESULTS TO FORMS OTHER THAN THOSE EXPERIMENTED 

UPON FACE FRICTION GEARING 

A fibrous driving wheel, acting upon the face of a metal disk, constitutes a 
form of friction gear which is serviceable for a variety of purposes. If the 
driver is so mounted that it may be moved across the face of the disk, the 
velocity ratio may be varied and the direction of the disk's motion may be 
reversed. The contact is not one of pure rolling. If the driver is cylindrical 
in form, the action along its line of contact with the disk is attended by slip, 
amount of which changes for every different point along the line. The recog- 
nition of this fact is essential to a discussion of the power- transmitting capacity 
of the device. 

Experiments involving the spur form of friction wheels already described 
have shown that slip greatly affects the coefficient of friction; that the co- 
efficient approaches its maximum value when the slip reaches 2 per cent., and 
that when the slip exceeds 3 per cent., the coefficient diminishes. It is know T n 
that reductions in the value of the coefficient with increments of slip beyond 
3 per cent, are at first gradual, although the characteristics of the testing 
machine have not permitted a definition of this relation for slip greater than 4 
per cent. The experiments, however, fully justify the statement that for 
maximum results the slippage should not be less than 2 per cent, nor more 
than 4 per cent. It is the maximum limit with which we are concerned in 
considering the amount of power which may be transmitted by face friction 
gearing. 



FRICTION GEARS 



3°3 



From the discussion of the previous paragraph, it should be evident that, for 
best results, the width of face of the friction driver and the distance between 
the driver and center of disk should always be such that the variations in the 
velocity of the particles of the disk having contact with the driver will not 
exceed 4 per cent. A convenient rule which, if followed, will secure this con- 
dition is to make the minimum distance between the driver and the center 
of the driven disk twelve times the width of the face of the driver. For exam- 
ple, a driver having a M-inch width of face should be run at a distance of 3 
inches or more from the center of the disk. Similarly, drivers having faces 
}4, 1, or 2 inches in width should be run at a distance from the center of the 
disk of not less than 6, 12, or 24 inches, respectively. When these conditions 
are met, all formulas for calculating the power which may be transmitted 
apply directly to the conditions of face driving. 

It may not infrequently happen that friction wheels must be run nearer the 
center of the disk than the distance specified; there is, of course, no objection 
to such practice, but it should not be forgotten that as the center of the disk 
is approached, the coefficient of friction, and consequently the capacity to 
transmit power, diminishes. 

CONDITIONS TO BE OBSERVED IN THE INSTALLATION OF 
FRICTION DRIVES 

Whatever may be the form of the transmission, the fibrous wheel must 
always be the driver. Neglect of this rule is likely to result in failure 
which will appear in the unequal wear of the softer wheel, occasioned by 
slippage. 

The rolling surfaces of the wheel should be kept clean. Ordinarily they 
should not be permitted to collect grease or oil, nor be exposed to excessive 
moisture. Where this cannot be prevented, a factor of safety should be pro- 
vided by making the wheels larger than normal for the power to be trans- 
mitted. 

Since the power transmitted is directly proportional to the pressure of con- 
tact, it is a matter of prime importance that the mechanical means employed 
in maintaining the contact be as nearly as possible inflexible. For example, 
arrangements of friction wheels which involve the maintenance of contact 
through the direct action of a spring have been found unsatisfactory, since 
any defect in the form of either wheel introduces vibrations which tend to 
impair the value of the arrangement. It is recommended that springs be 
avoided and that contact be secured through mechanism which is rigid and 
which when once adjusted shall be incapable of bringing about any release 
of the pressure to which it is set. 



304 AMERICAN MACHINIST GEAR BOOK 

EXPLANATION 01 CHART 

Chart 15 is plotted for the most common materials used for friction 
gearing, straw hber and cast iron, and gives means of determining the variable 
factors for the fiber wheel in the formula horse- power = 0.0003 <7TO • ho 
which d is the diameter of the wheel in inches, IT" its width of face in inches, 
and N the number of revolutions per minute. 

To use the chart for other friction materials multiply the values obtained 
from the chart by the proper factor selected from the table below: 

Straw fiber and alum in um 1 . 10 

Straw fiber and type metal o. 73 

Leather fiber and cast iron 1.97 

Leather fiber and aluminum 1 . 90 

Leather fiber and type metal 1.17 

Tarred fiber and cast iron 0.97 

Tarred fiber and aluminum 1.17 

Tarred fiber and type metal 1 . 03 

Sulphite fiber and cast iron : 1.23 

Sulphite fiber and aluminum 1.17 

Sulphite fiber and type metal 1 . 13 

Leather and cast iron 0.5; 

Leather and aluminum o. 87 

Leather and type metal o . 97 

a. To find the total horse- power which can be transmitted by a wheel, having 
given the diameter of the wheels in inches, the width of its face in inches and 
the revolutions per minute, locate the intersection of the vertical line represent- 
ing the given speed with the diagonal line representing the given diameter. 
Follow the horizontal line passing through this point, to the right or left as 
the case may be. until it intersects the vertical line representing the given 
width of face. The diagonal line through this point will give the total horse- 
power required from the scale so marked . 

b. To find the speed in revolutions per minute for a wheel, having given its 
diameter in inches, its width of face in inches, and the total horse-power to be 
transmitted, locate the intersection of the vertical line representing the width 
of face with the diagonal line representing the total horse-power to be trans- 
mitted. Follow the horizontal line passing through this point, to the right or 
left as the case may be. until it intersects the diagonal line representing the 
diameter in inches. The vertical line passing through this point indicates on 
the scale at the bottom of the chart the speed required. 



FRICTION GEARS 



3°5 



c. To find the width of face in inches for a wheel, having given the total 
horse-power to be transmitted, its diameter in inches and its speed in revolu- 
tions per minute, locate the intersection of the vertical line representing the 
given speed with the diagonal line representing the given diameter. Follow 



Face, Width in Inches 




5D C- 00 CS 



Speed, Revolutions per Minute 



CHART 15. PROPORTIONS OF FIBROUS FRICTION GEARING. 



the horizontal line passing through this point, to the right or left as the case 
may be, until it intersects the diagonal line representing the given total horse- 
power. The vertical line passing through this point will indicate the width 
of face required on the scale at the top of the chart. 

d. To find the diameter in inches for a wheel, having given the horse-power 
to be transmitted, its width of face in inches, and its speed in revolutions per 



306 AMERICAN MACHINIST GEAR BOOK 

minute, locate the intersection of the vertical line representing the width of 
face with the diagonal line indicating the total horse-power. Follow the hori- 
zontal line passing through this point, to the right or left as the case may be, 
until it intersects the vertical line representing the speed. The diagonal line 
passing through this point represents the diameter which is required. 

e. To find the surface speed of a wheel, having given its diameter in inches 
and its speed in revolutions per minute, locate the intersection of the vertical 
line representing the speed in revolutions per minute with the diagonal line 
representing the given diameter. The horizontal line passing through this 
point represents the surface speed in feet per minute which is required, and 
which is read on the vertical scale at the right of the chart. 

/. To find the horse-power per inch of face for a wheel, having given the total 
horse-power transmitted and the width of the face in inches, locate the inter- 
section of the vertical line representing the width of face with the diagonal 
line representing the total horse-power. The horizontal line passing through 
this point represents the horse-power per inch of face required and may be 
read on the vertical scale at the left of the chart. 

FRICTION DRIVE ON A FORTY-FOUR FOOT PIT LATHE* 

The machine here described was designed to meet the demands of an estab- 
lishment manufacturing the heaviest type of electrical machinery. The ever- 
increasing dimensions of this class of machinery make it particularly desirable 
that the existing heavy machine tools should be capable of extension of capac- 
ity with a view to probable future requirements, and that a pit lathe is 
peculiarly adapted to such extension will, doubtless, be readily admitted. 

The face-plate of this machine measures 30 feet in diameter, and the pres- 
ent dimensions of the pit will admit of swinging 44 feet on centers, with a 
maximum width of 12 feet. The large face-plate is built up of twelve seg- 
ments. The rim is of box section, the ends of the rim in each section being 
finished to make the joint, and the segments being held together at the rim 
by body-bound bolts. The arms are slotted for bolts, and the space between 
segments is also shaped to receive the usual square-headed bolts, as the inner 
end of each segment is fastened to the smaller face-plate by several body- 
bound bolts. 

A feature of interest in connection with this machine is the method of drive 
adopted, which is a friction roller, 18 inches diameter, made of compressed 
paper, while the rim of the large face-plate, 15 inches wide, affords the neces- 
sary contact surface for driving. 

* Extract from a paper presented at the New York meeting of the American Society of 
Mechanical Engineers by John M. Barney. 



FRICTION GEARS 



307 



Power is supplied by a 75 horse-power motor, quadruple-geared, the use 
of the multiple voltage system giving the machine a range covering all diam- 
eters from six feet to the present capacity, though the gear train is designed 
to admit of two changes of back gear in addition. 




FIG. 246. FRICTION-DRIVEN LATHE. 



Fig. 246 shows the assembled pit lathe driven by the friction roller while 
taking a heavy facing cut, on which occasion four tools were employed. The 
picture also shows the driving motor with its train of gears and the mechanism 
employed for adjusting the pressure on the friction roller. 



SECTION XIV 
Odd Gearing 

Under this head are shown a few examples of what has been produced 
in the way of odd gearing. 

THE GRISSON HIGH REDUCTION GEARING 

A good deal of interest has been manifested in Germany over the Grisson gear- 
ing for use in connection with electric motors, for which use a high ratio of 




FIG. 247. GRISSON GEARING, USING A THREE -TOOTHED PINION. 

speed reduction is very desirable. This gear is a revival in modified form of 
the two-toothed pinion, having the two teeth in different planes, the modi- 
fication consisting of the use of roller teeth in the larger gear. The two-toothed 

3©3 



ODD GEARING 



3°9 



pinion, with sliding contact and radial faces in the larger gear, is shown and 
discussed in McCord's "Kinematics," where the gear is said to be very satis- 
factory in operation except for the large amount of sliding which it entails. The 
use of roller teeth in the Grisson gear is, of course, to do away with this exces- 
sive sliding. 

Fig. 247 is a side elevation showing its appearance in an actual case. The 
pinion speed in the construction shown in Fig. 247 was 1200 revolutions per 
minute and the speed ratio was 12. The teeth of the pinion will be seen to be 
of the form of heart-shaped cams, each tooth working on its own set of roller 
teeth in the gear, these roller teeth being placed between appropriate flanges. 

The leading feature of the gear is, of course, its high ratio of speeds, together 
with great compactness and a small center distance. The smallest permissible 
ratio is said to be one to five and the 
action of the gears is better with higher 
ratios. The efficiency also increases 
with the ratio, experiments being said 
to have shown an efficiency as high as 
95 per cent. 

A NOVELTY IN GEARING 

The cuts show a somewhat novel style 
of gearing, the operation of which will 
be easily understood. The turning of 
the helical cam or pinion B moves the 
rack A j or in Fig 250 a wheel A is turned 
instead of operating the rack. The prin- 
ciple of the device is better shown in Fig. 
253, where the oblique continuous lines 
are avoided by making the gears in sec- 
tions, each successively a little in ad- 
vance of the preceding. This shows the 
driver B to consist of a series of eccen- 
trics, and of course the previous figures 
would show the same eccentricity in any 
section of the operative portion. The 
grooves of the rack, or of the driven 
wheel, should not be perfectly circular, 
but should be more nearly as shown in 
section in Fig. 252, with nearly straight lines at the points marked x. The 
specification of the patent for this gearing states: 




Fig. 253 
A NOVELTY IN GEARING. 



3io 



AMERICAN MACHINIST GEAR BOOK 




10 
10 



IS) 



ODD GEARING 31 1 

"The rotation of the member B imparts a movement to the member A, as 
a pinion will impart motion to a rack or wheel; but the member A is prevented 
from imparting any movement to the member B and is locked thereby when 
the parts are at rest, even more positively than a worm-wheel is locked by the 
worm." 

Patent was assigned to the Otis Elevator Co., 1901. 

TYPE CYLINDER GEAR 

Andrew Strom, of the Dayton Pneumatic Tool Company, Dayton, O., 
designed and cut the odd-shaped gear shown in Figs. 254 and 255, an ordinary 
milling machine being used. 

This gear is practically a round spur gear except for the two flat sides, and 
while it is not difficult to machine, it combines a spur gear and a rack in a 
way that is not pleasing to the gear cutting department of a shop. 




PUMP GEARS. 



The teeth are 0.6 pitch, pitch diameter 20.166, and a total of 120 teeth. 
The teeth were cut from the four quarters, with a tooth in the center and 21 
spaces each side on the round portions, and nine spaces each way on the flats. 



3 I2 



AMERICAN MACHINIST GEAR BOOK 





FIG. 257. FIG. 258. 

SPIRAL-TOOTH GEARS MADE BY CITROEN, HEUSTIN ET CLE, OF PARIS. 




FIG. 259. CROWN SPIRAL GEARS. 



ODD GEARING 



313 



▼ 





FIG. 260. FELLOWS SPUR BEVELS. 




FIG. 261. TWO FELLOWS SPUR-BEVEL GEABS 
IN COMBINATION WITH A SPUR GEAR. 



Bevel 



Bevel 



FIG. 262. TWO FELLOWS SPUR-BEVEL 
GEARS WITH PARALLEL AXES. 



Single Thread 




FIG. 263. COLLIER'S BALL-WORM GEAR. 



314 



AMERICAN MACHINIST GEAR BOOK 



This was cut by compound-index moves, set for 121 teeth, indexing from the 
two points for the teeth on the opposite side of the blank. Using both 47 and 
49 circle holes, move crank 1 \\ turns right hand and then backward 15 
holes in the 49 circle, or 1 if or — if* Or a special plate of 121 holes 
could be used, moving 40 holes for each tooth. 

In cutting the teeth on the gear for the impression cylinder, Fig. 25 5^, it was 
necessary to swing it from three centers, the main center for the regular por- 
tions of the gear and the two side centers for the smaller portions at each side, 
and the indexing became quite a difficult proposition. 




FIG. 264. A COLLIER DRIVE USING FOUR WORMS. 



The indexing was not continuous, but was started from four points, at the 
centers of the four arcs, with the tooth cut in the center, as shown. 

It took some careful calculating to find just what to do, but it was finally 
worked out to index in compound as follows: Three turns +25 holes in the 
47-hole circle, then 12 holes in the 49-hole circle in the same direction, giving 

3 47" + if f° r eacn tooth. 

After the center tooth was cut, 21 teeth were cut on each side of it, bringing 
the teeth to the beginning of the smaller radius. Then the blank was shifted 
to either end center and the center tooth was cut on this portion, then eight 
each way. This meets the other cutting, leaving the sharp-pointed tooth 
shown. 

This indexing is the same as for 60 teeth, but only 17 are cut in all. A No. 



ODD GEARING 



315 




Fig. 265 




AVorm Gear 



Spur Gear 



FIG. 266. ARRANGEMENT TO AVOID END THRUST IN A WORM DRIVE. 



316 AMERICAN MACHINIST GEAR BOOK 

2 cutter is used, being, of course, the same for all teeth, and is equivalent to 
a 6-pitch on the normal part of the gear. 

The indexing could also be done with a special plate having 233 holes and 
moving the index pin 80 holes for each cut. This worked out perfectly for 
this gear. 

Figures 256 to 266 are a collection of odd gear drives, giving the reader a 
faint idea of what it is possible to produce in this line. The arrangement of 
worm gears in Fig. 266 is often employed to counteract the end thrust of the 
worm shaft. 



SECTION XV 

Costs 

The foregoing sections have dealt exclusively with gears from the point of 
view of design, and with the assistance of the formulas, tables, etc., incor- 
porated throughout the discussions, the designer can plan and properly 
proportion any type of gear to meet efficiently all requirements for which 
the particular type is suited. It often happens, however, that gears are 
purchased and not built, and in such cases it is imperative that the designer 
order the gears with due regard to cost. There may be any number of dif- 
ferently proportioned gears of the same power capacity, many of which may 
also be efficiently proportioned so far as strength, etc., is concerned, but ob- 
viously all such gears cannot cost the same. The gear nearest standard pro- 
portions will almost invariably be the cheapest and the advisable one to 
employ. 

Standard proportions vary somewhat with different manufacturers, and 
many gears that are not of strictly standard proportions are quite com- 
monly stocked. This allows a more comprehensive choice on the part of 
the purchaser, but also necessitates more "cost knowledge" on his part. 
The commoner types of gears, the class of gears carried in stock by a manu- 
facturer, are pretty well standardized in the way of comparative cost. 
Individual prices may vary in different localities, but the relationship in 
cost of similar gears is fairly constant. This is particularly true for the 
commoner varieties of gears that are made up in quantities and stocked. 

SPUR GEARS 

A reliable relationship exists between the number of teeth of given pitch 
in a standard gear and its cost. This relationship varies proportionately 
with the number of teeth, but is different for each pitch. For instance, a 
cut, cast-iron gear of ioo teeth, i-inch pitch and 2 -inch face costs about four 
times as much as a similarly proportioned gear with only 25 teeth; while a 
cut, cast-iron gear of 100 teeth, 23^-inch pitch and 7-inch face costs about 
five times as much as a 25-toothed gear of similar proportions. 

The comparative costs — the cost of a 10-toothed, %-inch pitch, i-inch face 
spur gear being arbitrarily taken as unity and serving as the base of compari- 

3*7 



3i8 



AMERICAN MACHINIST GEAR BOOK 



COMPARATIVE COST 
40 SO 60 




4 -8 IE 16 20 24 28 32 36 40 44 46 52 56 60 64 68 72 76 80- 84 

COMPARATIVE COST 
BASE:- COST OF -g" PITCH, l"FACE, 10 -TOOTH 6EAR =1 

CHART 1 6. COMPARATIVE COST OF STANDARD CUT STEEL SPUR GEARS. 



10 



20 



COMPARATIVE COST 
30 40 50 60 70 



B0 



90 



100 




I 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 
COMPARATIVE COST 



92 96 100 



BASE:-COST OF j PITCH , |"FACE, 10 -TOOTH 6FAR =1= 1.34 X BASE FOR CHART 16 
CHART 17. COMPARATIVE COST OF STANDARD CUT CAST IRON SPUR GEARS. 



COSTS 



3 X 9 



son — of standard cut cast-iron and cut steel gears are graphically depicted 
on Charts 16 and 17, respectively. The standard proportions — circular 
pitch and face — of the gears are given in the vicinity of the curve to which 
they apply. These standard proportions are not adopted by all manufac- 
turers, nor are gears of exactly such dimensions always to be desired. 



SCALE FOR CAST IRON 6ETARS SAME AS FOR CHART 16 
>i »> STEEL. " " " " " |7 

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0.38 
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CHART 



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CIRCULAR PITCH 

1 8. CHARGE PER INCH INCREASE OF SPUR GEAR FACE TO BE ADDED TO 
COMPARATIVE COSTS, CHARTS 1 6 AND 1 7. 



An increase in the width of a standard gear adds to its cost, but this higher 
price depends upon the amount of the increase and the size of the gear — its 
diameter. For a given pitch, the extra cost depends upon the number of 
teeth. In fact there is a definite relationship governing the greater cost de- 
pending upon the amount of increase in the width of the gear and the number 
of teeth and the pitch. This is graphically depicted on Chart 18 for both 
cut cast-iron and steel gears. 

These comparative cost curves enable one to form quite reliable opinions 



320 



AMERICAN MACHINIST GEAR BOOK 



as to the respective costs of different gears, and though necessarily not ab- 
solutely accurate, do furnish the designer with information of value. 

For example, an installation might call for a 50-toothed, cut, cast-iron gear, 
2^-inch pitch by 8-inch face, a gear that would not vary much from stand- 
ard. The comparative cost of a gear of the same number of teeth and pitch 
but of 7-inch face, the standard, would be 15.1 (see Chart 16). The extra 
width charge would be 1 X 50 X 0.14 = 7 (see Chart 18), making the com- 
parative cost of the gear with 8-inch face 22.1. A 40-toothed, cut, cast- 
iron gear, 2)^-inch pitch by 73^-inch face, a standard gear, would have the 
same pitch diameter and have approximately the same power- transmit ting 
capacity. The comparative cost of such a gear (see Chart 16) would 
be 16.8, or the 234-inch pitch gear called for would cost nearly 33^ per 
cent, more than the standard 2^-inch pitch gear that might be substituted. 



















COMPARATIVE 

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COMPARATIVE C05T 
BASE--- COST OF !" PITCH, 2" FACE, IO-TOOTH GEAR -I 

CHART 19. COMPARATIVE COST OF STANDARD CUT CAST IRON BEVEL GEARS 



BEVEL GEARS 



Relationships similar to those for spur gears exist in the comparative costs 
of bevel gears (see Charts 19, 20 and 21). These records apply only to bevel 
or miter gears having 90-degree center angles, but also show the trend of 
prices for bevel gears of other but similar center angles. Any variation from 



COSTS 



321 



















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10 




COST 

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comparative cost 
base:- cost of ^" pitch, 2" face* 
10 -tooth gear =1 = 1.34 xbase for chart 10 

Chart 20. comparative cost of standard cut steel bevel gears,. 

SCALE FOR CAST IRON 6EARS SAME AS FOR CHART 19 
" " STEEL " m " » " ZO 



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CHART 21, 



CHARGE PER INCH INCREASE OF BEVEL GEAR FACE TO BE ADDED TO COMPARATIVE 
COSTS, CHARTS 1 9 AND 20. 



3 22 



AMERICAN MACHINIST GEAR BOOK 



a right-angle bevel installation usually entails added expense, however, so it 
is always advisable to use right-angle transmissions. 

Bevel gears of other angle may sometimes be obtained at as attractive 
prices as those for right-angle gears, but this cannot be counted upon. 
Furthermore, any variation from a 90-degree angle entails a sacrifice in 
mechanical efficiency. For both mechanical and economic efficiency it is 
well, therefore, to adhere as much as possible to 90-degree transmissions 
when employing bevel gears. 

WORM GEARS 

Worm gears are not in such general use as to be readily procurable in all 
sizes, but the comparative cost curves depicted on Chart 22 show the general 



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THE DIMENSION FOLLOWING THE 
CIRCULAR PITCH REFERS TO THE 
EXTREME FACE: WIDTH 




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125 
115 
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95 jj 

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85|" 
75 & 

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45 

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25 



BASE:- COST OF l" PITCH , S^EXTREME FACE WIDTH, 10 -TOOTH YV0RM6EAR = I 
CHART 2 2. COMPARATIVE COST OF STANDARD WORM GEAR WHEELS. 



trend of prices for this type of gearing. The curves are plotted from the cost 
of gears of similar pitch for each specific pitch and though the widths con- 
form to the standards of certain manufacturers, they are frequently varied. 
This would have a tendency to affect the comparative cost of such gears, but 
the relationships shown on the chart are typical, and as such should prove 
of assistance in the economic selection of suitable worm gears. 



COSTS 



323 







4 


8 


12 


16 


RATIO 
20 24 


IN 
28 


PER 
32 


CENT 

36 40 


44 


48 


52 


56 


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30 






























































































































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10 































































































































































































145 
35. 
125 
115 

105 

X 

95 fc 
ut 

85 H 
u_ 

750 

65 uj 

£0 

55 § 

z. 

45 
35 
25 
15 
5 



10 



To 



30 40 50 60 

RATIO BETWEEN COSTS OF WORMS AND WORM WHEELS 
IN PER CENT. 

CHART 23. PROPORTIONAL COST OF SINGLE THREAD WORMS — RATIO OF COST 

OF WORM TO COST OF WORM WHEEL. 



150 
140 

130 

120 

110 

1 100 

b 

u. 80 
o 

o: 70 
UI 

m 60 

z 

£ 50 
40 
30 
20 
10 























j 
































/ 
































/ 
































/ 
































/ 
































1 




























































































A/— 






























*y 






























S 




,£ 




























0} 




\U 


























^y~ 




w 


























\V 


<>?£ 






























A 






































rft, 


























y- 






f^/ 






























„?v 






























p.2 






































































































*fl 




























6jii 


?xk 

































































































































































































































b 



145 

135 

£5 

115 

105 r 
r* 

h 
85 u. 

O 

75 £ 

u 
65 CQ 

55 g 

45 
35 
25 
15 
5 



8 



3 4.5 

COMPARATIVE COST. 

BASE : ~ C05T OF 16 DIAMETRAL PITCH, 1 |" FACE., 10 -TOOTH SEAR* I 

CHART 24. COMPARATIVE COST OF STANDARD SPIRAL GEARS 45° SPIRAL 
ANGLE — SHAFTS AT 90°. 



324 AMERICAN MACHINIST GEAR BOOK 

The pinion cost is not included in that for worm gears, but the average 
proportional cost of single-thread worms suitable for use with the gears is 
graphically depicted on Chart 23, the proportional cost being the ratio 
of the cost of the worm to that of the worm gear in per cent. 

SPIRAL GEARS 

Spiral gears are not usually stocked except in small sizes. Chart 24 depicts 
the comparative cost of rather special spirals, in that the face is constant for 
a given pitch and the angle of spiral is the same for all gears. 

The gears are all small, hence the use of the diametral pitch in preference 
to the circular pitch, but the general trend of prices is clearly shown. The 
comparative costs for gears narrower or wider are very nearly directly 
proportioned to the face dimensions. 

Helical, herringbone and other special gears do not permit a reliable com- 
parative cost analysis to be made. They are nearly always made up specially 
and their cost depends upon the materials of which they are constructed and 
upon the facilities of the manufacturer. A well-proportioned gear, conform- 
ing as nearly as possible to general practice, is almost invariably the cheapest 
and most satisfactory. Freaks in gearing are always to be avoided. 



SECTION XVI 

Suggestions for Ordering Gears 

diameters 

When diameter is mentioned it is understood to be pitch diameter. The 
pitch diameter should always be given to check the pitch, as it often happens 
that diametral and circular pitch are confused. If the pitch diameter cannot 
be given give the outside diameter. 



Number of 

Teeth in 

Gear 

Material 




Number of 

Teeth in 

Pinion 



Material 



j<-Face->i !<-Hub 

Projection 

FIG. 292. NECESSARY DIMENSIONS TO COMPLETELY DESCRIBE A PAIR OP SPUR GEARS. 



HUBS 



Hub length and location should always be specified. If this is not given 
the usual practice has been to make the ends of hubs flush with face when the 
face is wide or the gear small in diameter. When the face is narrower and the 

325 



326 



AMERICAN MACHINIST GEAR BOOK 



gear large, it is common practice to make the hub longer than the face to give 
it bearing on the shaft. For the heavier class of gears this hub extension is 
generally made equal on each side of the face; for the smaller class, say 4 



Rotary Cutter / 




FIG. 293. CUTTING A SHROUDED GEAR. 



diametral pitch and under, the hub extension is generally put all on one side 
to accommodate set screws, etc. 

For pinions, especially those made of steel, it is always well to avoid hubs, 
as these add greatly to the cost, more, in fact, than if the face had been carried 
the length of the hub. 

When preparing blanks to be cut it is important that the ends of the hub 
be faced true, otherwise the blank will run out when clamped on the arbor. 

SHROUDED GEARS 

Gear manufacturers are often asked to furnish a shrouded cut gear. This 
is possible only by employing an end mill, which makes a very expensive 
operation. Where but one side is required to be shrouded, however, the cost 



SUGGESTIONS FOR ORDERING GEARS 



327 



will be somewhere within reason as the tooth may be milled part way by 
ordinary methods; the end milling being used for distance a in Fig. 293. 

Gears are quite often furnished (as shown in Fig. 293) without finishing dis- 
tance a, their face being limited to distance b. 




FIG. 294. NECESSARY DIMENSIONS TO COMPLETELY DESCRIBE A PAIR OF BEVEL GEARS. 



BORE 

The bore of a gear is generally made standard. Any allowance for a fit 
is understood to be made in the shaft. Gears sent to be cut should be bored, 
if possible, to some standard and be uniform in size, if extra charges for 
special bushings is to be avoided; or if the operator cutting the gears is of a 
saving nature and bushes up with a piece of tin, eccentric gears will be the 
result. 

All allowances for press, shrink, or sliding fits should be specified if you 
would expect satisfactory results, as there are various ideas on these points. 



328 AMERICAN MACHINIST GEAR BOOK 

The taper bore is rapidly coming into favor, as it is easier to machine with 
proper methods and insures a snug tit on the shaft and a true gear running. 

KEYS 

Unless otherwise specified, keyways are understood to be straight. If 
taper, the size given is at the small end. See pages 130 and 134 for standards. 

Where hubs are unequal, or in the case of bevel gears, specify the direction 
from which key is to drive, although it is understood to be from the small end 
of a bevel gear unless otherwise specified. When two keys are required place 
them diametrically opposite instead of at 90 degrees. 

Avoid the use of a taper keyseat; they will cause the gear to run out of true 
even if the bore is a snug fit unless extreme care is taken. 

BEVEL GEARS 

As a great many do not understand what information is necessary in describ- 
ing a pair of bevel gears, and are unable to make up drawings the following 
list is given, which, if properly filled out, will be all that is necessary, either 
for a new transmission or to replace worn gears. Fig. 294 illustrates the 
corresponding dimensions. 

Gear 

Number of teeth N 

Pitch p' 

Face b 

Bore 

Pitch diameter D' 

Outside diameter D 

Backing Z 

From point of tooth to center of pinion shaft S 

Length of hub L 

Diameter of hub H 

Keyseat 

Material 

Pinion 

Number of teeth n 

Pitch p' 

Face b 

Bore 

Pitch diameter. d f 



SUGGESTIONS FOR ORDERING GEARS 329 

Outside diameter d 

Backing z 

From point of tooth to center of gear shaft s 

Length of hub I 

Diameter of hub h 

Keyseat 

Material 

In addition to these dimensions it is advisable to send a paper impression 
of the teeth, if replacing old gears; this can be made by laying a piece of clean 
paper over the large end of the teeth, which are first slightly greased or lamp- 
blacked. Then crease around the edges of the teeth with the end of a pencil. 
This is very necessary unless the pitch can be given, which is the distance from 
the center or edge of one tooth to the center or edge of the next — not measured 
at the points, but on the pitch line. 

It is not necessary to make a sketch unless the gears have some special 
feature, simply give the list of dimensions and the paper templet as described; 
although a sketch is always best, as many points may come up that would 
otherwise have been overlooked. When ordering one gear of a pair always give 
the number of teeth in its mate. 

When the ends of the hubs are against a bearing, or collar arranged for that 
purpose, the distance 5 and s should be accurately determined. If, however, 
there is some distance between the ends of hubs and the bearing, these dimen- 
sions are of no special importance. It is good practice to allow an extra 
H inch on the ends of the hubs, for fitting when this dimension must be accu- 
rate. State whether keyway is straight or tapered, and if tapered, from 
which side it drives. 

It is assumed that the shafts are at right angles unless otherwise specified. 
When such is not the case the only additional information needed is that shaft 
angle, although the distance S and 5 should be from their point of intersection. 
These instructions also apply to miter gears. 

The pitch diameter D' and d f in Fig. 294 should be given when it is possible. 
The pitch diameters are generally specified for new work, but Fig. 294 and 
accompanying instructions were made up in such a manner that they could be 
used to order replacements, therefore the outside diameter is shown and should 
be given in case the pitch diameter cannot be determined. 

Always give all information possible when ordering replaced bevel gears on 
account of various corrections now made in the tooth dimensions. 

Determine if any corrections in angles will be required before turning blanks 
to be cut, as some correction is necessary for all generated bevel gears of 
143^ or 15 degrees. 



33° 



AMERICAN MACHINIST GEAR BOOK 



It is often just as cheap or cheaper to have a small hub on the back of the 
bevel gear if made from a casting. 

When no backing is specified the manufacturer will make same to suit his 
ideas, which vary. Backing should always be given so that the gears may 
seat against bearing; end of bearing should be finished for this purpose. 

Avoid putting long hubs on front end of gear as they are often impossible 
to cut. 

If you turn up your own gear, be sure that the distance from the point of the 
tooth to the end of the hub (backing) is the same in all of them. If this dis- 
tance is not maintained it adds greatly to the cost of cutting. As some genera- 
ting machines hold the gear from the small end, it is important, if they are to 
be cut on this type of machine, that the distance from the point of the tooth 
to the front end of hub be uniform. 

If you do not understand how to turn up bevel gear blanks, send them to the 
manufacturer as it is expensive experience for you — also for the manufacturer — 
to attempt to cut gears incorrectly machined. This applies to all types of 
gears. 

If the six arms or ribs are used in bevel sears large enough to be turned on 
boring mill, it will greatly reduce the turning time if small lugs are cast on the 
bottom of rim to engage chuck jaws, which are generally four in number. 
This is especially important if the ribs are extended so that the chuck jaws 
cannot grip the hub. 

When bevel, or for that matter any other type of gears are made from a 
casting, it pays to add a small hub, if none is required, by which to hold the 
gears in the chuck; the labor turning oft this hub is small in comparison to the 
time saved. 

WORM GEARS 

Before designing a worm gear it should first be determined whether a hob 
is obtainable that will cut the gear. Gear manufacturers will furnish such a 
list from which one should be selected. 

When ordering a worm gear always specify what amount, if any, that centers 
can be shifted. 

Worms should be keyseated before the thread is mi lled or chased if the cost 
is of importance. A bronze rim worm gear and a hardened steel worm is a 
good combination, but a hardened steel worm gear engaging a hardened steel 
worm is better. 

Worm gears should be carefully ground after being assembled with light oil 
and graphite, otherwise they are very liable to wear quickly. 

It is not necessary to turn out the throat radius of worm gear; a single spot 
in center of throat turned to the throat diameter will answer. 



SUGGESTIONS FOR ORDERING GEARS 



331 



Number of 
Teeth 



Material 



Hand. 
Lead. 
(Material. 



I<r— 



|<e Length of Thread -^M Projection 
—Length — >| 




K— Outfeide ->| 
J Piametejj \ 



FIG. 295. NECESSARY DIMENSIONS TO DESCRIBE WORM GEARS. 
SPIRAL GEARS 

Whenever it is possible, have calculations for spiral gear made before the 
center distance is decided upon. Specify maximum or minimum diameters 
and center distance. 

Pitch mentioned in connection with spiral or helical gear is understood to be 
the normal pitch, which, in circular pitch, is the shortest distance between two 
consecutive teeth. 

The gear with the greatest angle must be the driver. 



RACKS 

Avoid ordering racks of an odd thickness as it often requires months to 
secure special sizes in cold rolled steel. The thickness over all should be in 
the nearest even sixteenth of an inch; if this is impractical, order rack made of 
forged steel planed all over. It is not practical to plane up a piece of cold 
rolled steel. Before designing a spiral rack make sure of the maximum angles 
that can be cut. 



332 



AMERICAN MACHINIST GEAR BOOK 




wmmmm 



T 

— 



: — 



FIG. 296. INTERNAL GEAR ANT) PINION. 



INTERNAL GEARS 

If a rotary cutter must be used in cutting internal gear, they must be de- 
signed according to Fig. 297, which will be self-explanatory. A special cutter 
is generally required, but for 4 diametral pitch and finer and for 60 teeth and 

over, a regular number 1 cutter will 
make a very satisfactory job. 

The Fellows gear shaper will cut in- 
ternal gear with a minimum amount of 




Clearance 




FIG. 297. CUTTING INTERNAL GEAR WITH 
ROTARY CUTTER. 



EIG. 298. CLEARANCE FOR INTERNAL 
GEARS. 



SUGGESTIONS FOR ORDERING GEARS 



333 



clearance ( T Vrnch). Care must be taken, however, that there is room for the 
cutter between the inside of the rim and the outside of the hub as illustrated 
in Fig. 298. 

There should be at least 15 teeth difference between the number of teeth 
in the internal gear and the number of teeth in the engaging spur gear. 



Internal Gear Ring 
30 Teeth 




FIG. 299. SPECIAL INTERNAL GEAR DRIVE. 



This same rule applies to the relation of the numbers of teeth in the internal 
gear and the number of teeth in the Fellows cutter. 

If the difference between the two is less than 15, the points of the teeth in 
internal gear will be cut away. 



334 AMERICAN MACHINIST GEAR BOOK 

When it is necessary to design an internal gear drive and the difference in the 
number of teeth is slight the teeth can best be laid out by the cycloidal system, 
the diameter of the describing circle (see page 3) being made equal to the pitch 
diameter of one-half the difference in the number of teeth (see Fig. 299). 

Fig. 299 shows a case where there is but one tooth difference between the 
internal gear and the engaging spur gear. 

INTERCHANGEABILITY OF PARTS 

If any attempt at interchangeability of parts be desired, put limitations on 
drawings, plus and minus. Mark parts to be left rough, or not necessarily 
accurate. Give manufacturer idea of how parts are assembled. Send assem- 
bled drawing, or, still better, a sample set of gears and engaging parts. Send 
parts to be fitted, especially if of foreign make. 

SET SCREWS 

Do not locate the set screw so far under the rim of the gear that it will be 
necessary to drill hole with a ratchet. If a key is also used, locate set screw 
over key. 

PINS 

A taper pin is a poor means of securing the gear, unless used as a safety 
appliance to prevent a more serious break. .It falls out and damages other 
parts, unless secured by a nut on small end; and shear, unless made unpro- 
portionately large — a key is better. 

MATERIAL 

When no material is specified manufacturers will ordinarily understand that 
cast iron is wanted. 

It is generally advisable to make small pinion of machine steel for obvious 
reasons ; also a steel pinion is often as cheap, if not cheaper, than one of cast iron. 
This applies also when a large number is required on account of the superior 
facilities for handling and machining bar steel, especially in automatic machines. 

Where machine or forged steel is specified it is understood that a steel ap- 
proximately 0.30 carbon is wanted. The next higher commercial grade 
approximates 0.50 carbon. It is not advisable to use a higher carbon content 
than 0.50 for gears on account of the tendency of the higher carbons to crystal- 
lize in service. Although it is sometimes permissible to use high carbon steel 
or spiral gears (the action of the teeth preventing shocks which tend to crystal- 
lize the material), a carbon content as high as 1.20 is often used with success 
for such gears. 

Gears for case hardening are generally made of steel lower than 0.20 carbon, 



SUGGESTIONS FOR ORDERING GEARS 



335 



12 to 14 being recommended by many. If any other grade is required it 
should be specified. The customer should also state whether cyanide or bone 
hardening is required. To obtain a harder as well as a tougher gear that will 
withstand hard usage and heavy loads, it is necessary to resort to an alloy 
steel which may be tempered. The most commonly used is a steel containing 
about 0.30 carbon and 3^ -per cent, of nickel. Chrome nickel or chrome 
vanadium steel is much used in high-class automobile gears. There are many 
grades of alloy steels, some for case hardening, some for oil or water tempering; 
owing to their great variety and varying values, however, no attempt is made 
to enter further into this subject. 

When gears are to be case hardened or tempered there are several points 
that should be kept in mind. No machine work is possible on hardened gears, 
except by grinding; therefore, for parts that must be exact size, it should be 
specified what allowance be made for this, as the gear is liable to either expand 
or contract during the process of hardening, due to either the condition of the 
steel before machining, the general construction of the gear, or to strains set 
up by the removal of a considerable stock. For this reason it is well to first 
be sure that the blank is properly annealed, and that if any amount of stock 
is removed, and it is necessary to maintain accurate sizes in the finished gear, 
the roughed out blank should again be annealed before finishing; a gear 
with a square bore is an example of this. 

It is always a safe plan to have steel castings and drop forgings annealed. 
This is especially important with drop forgings, as they are generally allowed 
to cool from the forging heat on the dirt floor (dies are often hardened this 
way), and the process of forging is sure to set up internal strains no matter what 
their shape. 

Pin holes through the hub must be drilled before gear is hardened, unless the 
consumer is to do the hardening. Case hardening gears are often simply 
carbonized, that is, packed in bone and subjected to heat for the proper time, 
but not hardened. This extra heating tends to make a better gear than when 
they are dipped direct from the carbonizing pot ; also it offers an opportunity 
to finish up parts which must be close to size, as the blank has been relieved of 
strains. If necessary, portions may be machined beyond the depth to where 
the carbon has penetrated, leaving that surface soft after machining. To do 
this, however, the carbon content must be below 20 per cent., as the higher 
carbon steels will harden throughout. 

For larger gears the teeth are often cut in a rolled steel rim which is keyed 
or shrunk on a steel center, as the best steel casting often falls short of meet- 
ing the required conditions. This type is used for street railway gears for 
heavy service (see page 126). 



336 AMERICAN MACHINIST GEAR BOOK 

No definite rule can be laid down for the shrinkage of tempered gears, 
although it is reasonably well established that properly treated alloy steels 
shrink less than ordinary carbon steel. Each particular grade of steel and 
style of gear must be tried out separately. An approximate rule, however, 
would be to allow 0.0005 per inch of diameter. Some gears will require more 
than this and some less according to their design. A uniform section of 
material throughout the gear will tend to minimize these distortions. 

WORKING DRAWINGS FOR GEARS 

If it was realized just how much a proper drawing will facilitate production, 
or rather the delays that are caused by the lack of such drawings, there would 
be a decided change from the present practice. Many draftsmen seem 
to see just how many gears they can crowd on to one sheet; the gears are 
shown in mesh, and working dimensions are conspicuous by their absence. 
Sometimes a blank sketch is shown with a long list of dimensions covering 
the balance of the print ; the gears listed may be made of three or four different 
kinds of materials. This is all very nice from the draftsman's standpoint, 
but how about the workman who has to trace out all of the necessary di- 
mensions? The pitch diameter and backing of bevel gears from the pitch 
line are necessary dimensions — for the draftsman — but they are of no in- 
terest to the workman. . Some one must put these drawings in shape before 
they go into the shop; perhaps new ones must be made. This takes time and 
means a delay right at the very beginning of the job, and the more difficult 
the work the longer the delay, for more than one reason. 

The revised print finally reaches the shop. Suppose there are a dozen 
patterns to be made and the gears are all on one print, which also shows several 
forged steel gears. No work can be done on the forgings until the pattern 
maker is through with the print; the time taken to construct and check the 
patterns is lost as far as the steel gears are concerned, which otherwise might 
have been completed. When the machine work is started the print must 
follow each gear, in the meantime the balance of the work is waiting unless the 
foreman is a rapid draftsman. This process must be repeated in each depart- 
ment, and yet the customer wonders " where are his gears. " If his draftsman 
had spent some of the time used in drawing standard gear teeth — to which no 
attention is paid — on essential dimensions, the work would have been further 
along. A description of the style of tooth is all that is necessary. Unless a 
special form of tooth is desired, it is best to show none at all, as this will cause 
another delay until it is ascertained what is required. 

A drawing should show all necessary working dimensions. No figuring should 
be required of the workman into whose hands it is placed. 



SUGGESTIONS FOR ORDERING GEARS 



337 



When special material or special treatments are to be used, always put this 
information in full on the drawing; also give working limits, and specify sur- 
faces to be ground as well as those upon which no finish is required. It is 
often advisable to show adjacent parts in dotted lines, so that better judgment 




J. 




b%" Eough turn. 




"p. dia— J | -40 Teeth 6 Pitch. Pinion 14 teeth, B P 318 A. 

8.126- 



Center and back angles 71 34' 

Face angle 15° 51' 

Cutting angle 68° 36' 

Tooth angle 2o°oo' 

3H% nickel steel case hardened in oil 



Addendum .0.250" 

Whole depth 0.785" 

Thickness 0.380" 



FIG. 3OO. DIMENSIONED BEVEL GEAR. 



be used in the machine work. And above all other things, put the gears upon 
separate drawings, especially if made in quantities, as a drawing cannot be 
in two places in one time; when working upon one gear, a workman is not 
interested in its mate; besides, this makes the drawing plainer. This applies 
to all types of gears. Fig. 300 is an illustration. 



SECTION XVII 

Practical Points in Gear Cutting 

The outside diameter of the gear blank should first be measured and any 
variations noted, so that the proper allowance can be made in the tooth 
dimensions. 

All burs should be removed from the ends of hubs which should be faced true 
with bore. 

When putting several gears together on work arbor they must be faced 
unusually true or the arbor will be bent. This is especially true when the bore 
is small and the gears are large in diameter. 

Be sure that work arbor has a proper fit in spindle of machine before tight- 
ening the draw bolt, also that it runs true before putting on the gear blank. 

If a stock bushing does not fit the bore have one made. Do not use paper 
or tin; and do not use too many bushings. 

Be sure that the proper side of rim goes against the back steady rest; this 
side should be chalked before the gear is taken out of the chuck when turning 
up the blank. 

Use some judgment in tightening the nut on work arbor; that is, do not use 
or hammer on an 1 8-inch wrench for a one-inch arbor. 

If outer support is used, be sure that it holds work arbor in a horizontal 
position. Special attention must be given to this point when mounting heavy 
gears, as they are very likely to draw down the outer end of arbor. 

Throw out index worm and make sure that the gear runs true. 

When setting steady rest be sure that there are no obstructions on the rim 
that will strike it and spoil the gear; also oil the surface of rim. 

Put on proper feeds and speeds to suit the grade of material being cut and 
the type of cutter used. 

Nick the gear around before starting the cut to prove the indexing. 

Be sure that finishing cutter is central. 

When dropping cutter for depth of tooth, allow for any error in outside diam- 
eter, that the tooth may be of the proper thickness at the pitch line. 

Finish just enough of the end of one tooth to be sure of your thickness before 
proceeding further. 

Use chordal tooth parts for measuring the teeth; do not depend upon the 

shake of the tooth gauge. 

338 



PRACTICAL POINTS IN GEAR CUTTING 339 

If the pitch is not too coarse, two cutters may be employed; one for roughing 
and one to finish, the roughing cutter being separated from the finishing cutter 
by a spacing washer. 

Care should be taken that the rougher cutter is not too large in diameter or 
too wide. If the teeth are first roughed out, the roughing cutter should be 
made to cut the full depth, leaving no stock for the finishing cutter at the 
bottom of tooth. 

Roughing cutter should make central cut in tooth space, otherwise one side 
of the finishing cutter will wear rapidly. 

If the cutter shows signs of distress try cutting down the speed before chang- 
ing the feed per minute. 

Always use two cutters when possible. It has been argued that this makes 
inaccurate gears, owing to the change in temperature between the start and 
finish of the cutting; but such is not the case. 

Do not remove gears from the machine until tooth thickness all around has 
been inspected; this, of course, is impractical for small work. 

Use plenty of lubricant when cutting steel. If soda water does not seem to 
answer use oil; on the other hand, if oil does not give results, try soda water. 
When cutting bronze try cutting it dry if a lubricant does not give results, and 
vice versa. 

Be sure that the cutting edges of finishing cutter are ground radial, other- 
wise an incorrectly shaped tooth will be the result. 

Be sure that the key driving cutter does not bind on the top, otherwise the 
cutter will be broken. There are generally fillets in the corners of the cutter 
keyseat, therefore the tops of key should be beveled off to suit. 

Do not grind the cutter to suit the arbor; if the arbor runs out of true have 
it corrected — it will pay in the end. 

It pays to use high-speed steel cutters, especially for the finer pitch (4 dia- 
metral pitch and under), but put up the speed to at least double that used 
for carbon cutters, keeping the speed per revolution of cutter the same. 

Remember that it is not possible to operate a gear cut absolutely without 
backlash — there must be some allowance made. When it is known how the 
gears are to engage it is suggested that the pinions be cut standard; that is, no 
allowance be made, and that the teeth in engaging gear be made 0.01 of the 
circular pitch thin on the pitch line. On the other hand, when it is not defi- 
nitely known how the gears engage cut all teeth 0.005 °f the circular pitch thin. 
This would seem a better rule for general use. When generating bevel gears 
it is common practice to make the pinion tooth heavier than the gear tooth 
(according to the ratio), to give additional life to the pinion. Special pro- 
vision is made on most bevel generating machines to make this setting. 



34Q 



AMERICAN MACHINIST GEAR BOOK 




-Centers 



American Machinist 



FIG. 3OI. A SIMPLE TESTING FLXTURE. 



When the number of gears to be cut justifies it, have pair of pins turned to 
fit the bore of the gears set in a plate at the proper center distance. This plate 
is to be kept at the machine to test the work, as the cutter must be readjusted 

as it wears and changes in diame- 
ter. Exact centers on their jig may 
be cheaply attained by turning the 
stub of one of the pins entering plate 
eccentric, pinning it securely when 
adjusted to the proper centers (see 
Fig. 301). 

After gears are cut there are always 
slight tool marks, burs, and minor 
inaccuracies that will prevent their 
smooth operation when first assembled. If facilities are at hand to give 
the gears a running test, case-harden one gear out of the lot and run the 
balance of the gears with it for a short period, in both directions. This will 
remove tool marks, insure the smooth operating of the gears, and facili- 
tate assembling of parts. 

For spur gears it might be sug- 
gested that a gear for this purpose 
be cut on the Fellows shaper, 
or similar generating machine, en- 
larging the diameter of the gear 
until the pitch diameter is near the 
bottom of the tooth. 
N+4 



Pitch Diameter 



D' = 




P 



American Machinist 



FIG. 302. ARRANGEMENT OF PITCH DIAME- 
TERS TO POLISH TEETH BY SLIDING. 



The teeth of a gear cut in this 

manner will have no rolling action, 

the pitch diameters not touching (see Fig. 302). The resulting sliding 

action is just what is required for the purpose mentioned — that is, polishing 

the teeth. 



INDEX 



Addendum and dedendum angles of bevel 

gears, 148-151 
Anglemeter, 98-106 
Arc of tooth contact, 29 
Arms, hollow, 125 

I-shaped, 117 

number of, 113 

spur gear, 1 21-125 
Automobile worm drives, 195-196 

cost, 196 



Bevel gears, 139-162 

addendum and dedendum angles, 
148-15 1 

angles, 139-142 

chart, use of, 147 

comparative cost, chart, 320-321 

diameter and angles, 148 

efficiency, 160 

formulas, 144, 145-146 

generating machines, 159 

hardening, 160-161 

layout, 139-142 

machines for cutting, 153 

machining, 153-160 

milling, 154-159 

notation, 143, 145 

ordering, 325-337 

parallel depth, 155 

proportions, 127 

strength, 65, 67, 68 

table, 152 

use of table, 146 
Bilgram machine, 8 
Bore of gears, 134 
Buttressed teeth, 30 

Cast iron shrouded teeth, 73 

cut gears, 64 
Chordal pitch, 39 

tooth thicknesses, 41-43 
Circular pitch table, 34 



table, 



Classifications of gears, 2 
Collier's ball worm gear, 313 
Comparative costs, bevel gears, 320 

spiral gears, 324 

spur gears, 317 

worm gears, 322 
Connecting rod arm, 118 
Cored teeth, spur gears, in 
Costs, 317-324 
Crown spiral gears, 311 
Cutters, involute, 44 
Cutting, practical points, 338-340 

DeLaval speed reduction gear, 210 

Design, details of, 1 10-138 

Diagram, involute gear interference lo« 

cation, 18 
Diametral pitch table, 33 

pitch worms, 203 
Differential gearing, 272-277 
Drawings of gears, 336 

Efficiency, bevel gears, 160 

gears lubricated, 105 

gears without lubrication, 105 

herringbone gears, 223 

involute, 106 

large gears, 107 

worm gears, 1 81-194 
Elliptical gears, 265-271 

cutting, 267 

interference, 271 

laying out, 265-266 

pitch lines, 270 
End thrust, counteracting, 315 
Epicyclic gear trains, 272-287 

calculations, 272 

differential back gears, 286 

direction of rotation, 279 

finding velocity ratio, 278 

idler, 275 

internal gears, 275-277 

velocity ratio formulas, 272-278 



34i 



342 



INDEX 



Face width, 114-115 
Fellows shaper, 9 

spur bevels, 313 
Fixture, testing for cutting, 440 
Force fits, 136-137 
Formulas, bevel gears, 144, 145-146 

epicyclic gear trains, 272-278 

gear proportions, 111-112 

helical and herringbone gears, 206-208 

reversible worm and gear, 168 

skew bevel gears, 248-250 

spiral gears, 233-236 

spur gear calculations, 45-48 

spur gear interference point, 17 

strength of teeth, 76-77 

worm gears, 164-167 
Friction gears, 288-307 

applications, 302-303 

chart for, 305 

coefficients, 293, 299, 300-301 

fibrous wheels, 298 

horse-power, 301-302 

materials used for, 288 

metal wheels, 297 

pit lathe drive, 306 

power capacity, 298 

resistance to crushing, 297 

strength, fibre, wheels, 299 

testing, 289-293 

working pressures, 300 

Gear trains, arrangement, 50-51 
Generating machines for bevel gears, 159 
Grinding bevel gears, 161-162 
Grisson high-reduction gearing, 308 

Hardening bevel gears, 1 60-1 61 

Hardening gears, 335 

Helical and herringbone gears, 204-224 

advantages and applications, 218-220 

formulas, 206-208 

notation, 206 
Helical gears, design, 205 

examples, 208-209, 210 
Herringbone gears, 21 1-2 12 

advantages, 218 

determining lead from sample, 221-223 

efficiency and strength, 223 



Herringbone gears, formulas, 206-208 

hobbing process, 214 

important points, 220 

interchangeable system, 212-213 

machining, 213-218 

milling process, 216-218 

modified, 223-224 

planing process, 216 
High-speed gearing, 91-98 
Hindley worm gear, 197-203 

making, 199-203 

strength, 198 
Hoist gearing, 53-55 
Hollow arms, spur gears, 125 
Hub diameters, outside, 113 
Hunting tooth, 31 

Intermittent gears, 254-264 

Geneva stop, modifications, 256-260 

spiral, 262-264 

spur, 261 

worm, 261 
Internal gear and pinion interference, 22 

gears, ordering, 332-333 
Involute cutters, 44 

drawing curve, 10 

fifteen degree tooth, 60-63 

gear, diagram, location of interference, 18 

gears, interference, 17 

model demonstration, 11-15 

origin of, 7 
I-shaped arms, 117 

Key, Kennedy, 134 
Keyseats, 131 
Keys, ordering, 328 
Woodruff, 132-133 

Lanchester worm gear efficiency, 194 
Law of tooth contact, 28 
Loads, safe working, table, 64 
Location, influence on design and speed, 89 

Machines for cutting bevel gears, 153 
Machining, bevel gears, 154-159 

herringbone gears, 213-218 

skew bevel gears, 251 
Materials for gears, 334 



INDEX 



343 



Metric pitch, 38 
Milling bevel gears, 154-159 
Mitre gears, 142-143 
Modified, involute teeth, 26 
Mortise gears, 130 

Octoid tooth, 3, 27 
Odd gearing, 308-316 

Collier's ball worm, 313 

counteracting end thrust, 315 

crown spirals, 312 

Grisson, high-reduction, 308 

pump gears, 311 
Ordering gears, 325-337 

bevel, 328-330 

internal, 332-333 

spur, 325 

worm, 330-331 
Origin, involute tooth, 7 

Parallel depth bevel gears, 155 
Pins, fastening, 334 
Pitch, chordal, 39 

circular, table, 34 

diameters, chordal, table, 40 

diameters, circular, table, 38 

diametral, table, 33 

metric, 38 
Pitches, definition, diametral and circular, 

32 
Pressure, relation to speed, 90 
Pressures, fit, 135 
Proportions and design, 1 10-138 

arms, number of, 113 

bevel gears, 127 

connecting-rod arm, 118 

cored teeth, in 

face width, 114-115 

formulas, 111-112 

hollow arms, 125 

hub diameters, outside, 113 

I-shaped arms, 117 

key, Kennedy, 134 

key, Woodruff, 132-133 

keyseats, 131 

mortise gears, 130 

rawhide gears, 128-129 

rim gears, 126 



Proportions and design, skew bevel gears, 
245-246, 251-252 

split spur gears, n 6-1 17 

spur gears, no 

webbed spur gears, 115 
Pump gears, 311 

Rack and pinion interference, 20 
Racks, ordering, 331 
Railway gears, 56 
Ratio of power, 51 
Ratio of speeds, 49 
Rawhide gears, 128-129 
Reversible worm gears, 168 
Rim gear proportions, 126 

Safety, factor of, 65 

Set screws, 334 

Shrouded gears, cutting, 326 

teeth, 68 
Sizes of teeth, involute, 34-37 
Skew bevel gears, 245-253 

design, 245-246, 251-252 

example in design, 250-251 

formulas, 248-250 

machining, 251 

notation, 247 
Speed and powers, 49-109 

anglemeter-measuring, 98-106 

bevel gears, 67, 68 

cast iron shrouded teeth, 73 

construction of gear, 89 

cycloid and fifteen degree involute, 60-63 

determining teeth in contact, 88 

efficiency, involute, 106 

efficiency, large gears, 107 

factor of safety, 65 

high-speed gearing, 91-98 

hoist gear example, 53-55 

influence of location, 89 

limiting speeds, 88 

loads, safe working, table, 64 

pressure, relation of, 90 

railway gears, 56 

shrouded teeth, 68 

spur gears, safe load, 82 

strength of teeth, 57-58 

tooth strength and hardness, 84-85 



344 

Speed limits, 88 

Speed of gears, 49 

Speed reduction gear, DeLaval, 210 

Spiral gears, 225-244 

angles, 226 

cutters, chart, 243 

formulas, 233-236 

laying out, chart, 237 

notation, 231-232 

rotation and thrust, 241 

speed ratio, 229 

table, 239 
Spur gear arms, 1 21-12 6 

circular pitch, 48 

connecting-rod arm, 118 

diametral pitch, 47 

gear, calculations, 45-48 

interference point formula, 17 

mating formulas, 46 

ordering, 325 

proportions, no 

safe load, 82 

speeds, 86-87 
Spur gears, split, 116-117 

webbed, 115 
Stepped gears, 31 
Straight cut worm gears, 179 
Strength of teeth, 57-58 

Templet making, 31 

Thickness of tooth, chordal, 41-43 

Tooth, buttressed, 30 

contact, 88 

contact law, 28 

cycloidal, 3-4 

design, 86 

hunting, 31 

involute, 3, 8-9 

modified herringbone, 223-224 

octoid, 3 

single curve, laying out, 10 



INDEX 



Tooth, standards, 23-26 

author's, 23 

Brown & Sharpe, 23 

Fellow's, 24 

Grant's, 23 

Hunt's, 23 

Sellers', 23 

universal, 25-26 
strength, 57-58 
strength and hardness, 84-85 
thickness, chordal, 41-43 

Webbed spur gears, 115 

Weight, cast iron gearing, n 9-1 21 

steel gearing, 120 
Worm gears, 163-203 

arrangement of teeth on a worm-gear hob, 
170-176 

cutting on milling machine, 178 

diametral pitch, 203 

efficiency, 181-192 
and temperature, 193 
chart, 184 

formulas, 165-167 

Hindley worm wheel, 197-203 
calculating strength, 198 
making, 197-203 

hob, 169 

Lanchester, efficiency, 194 

limiting speeds and pressures, 186 

manufacturing processes, 1 78-181 

materials, 180 

notation, 164 

number of flutes to hob, 170-176 

ordering, 330-331 

power and efficiency, 181- 194 

reducing diameters, 177 

relieving spiral fluted hob, 176-177 

reversible, 168 

speeds and pressures, 192 

straight cut, 179 



